असमानता \(3x+4\ge -2\) का संख्या रेखा पर सही निरूपण क्या है?
What is the correct number line representation of \(3x+4\ge -2\)?
#linear-inequalities
#closed-circle
#solving
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A (-2) पर खाली वृत्त और दाईं ओर / Open circle at (-2) and right side
B (-2) पर भरा वृत्त और दाईं ओर / Closed circle at (-2) and right side
C (-2) पर भरा वृत्त और बाईं ओर / Closed circle at (-2) and left side
D (-2) पर खाली वृत्त और बाईं ओर / Open circle at (-2) and left side
Explanation opens after your attempt
Correct Answer
B. (-2) पर भरा वृत्त और दाईं ओर / Closed circle at (-2) and right side
Step 1
Concept
Solving gives \(x\ge -2\). Because of \(\ge\), the boundary is included and shading goes right.
Step 2
Why this answer is correct
The correct answer is B. (-2) पर भरा वृत्त और दाईं ओर / Closed circle at (-2) and right side. Solving gives \(x\ge -2\). Because of \(\ge\), the boundary is included and shading goes right.
Step 3
Exam Tip
हल करने पर \(x\ge -2\) मिलता है। \(\ge\) के कारण सीमा शामिल होती है और दाईं ओर छायांकन होगा।
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\(4-3x\le 10\) को संख्या रेखा पर कैसे दिखाएँगे?
How will \(4-3x\le 10\) be shown on the number line?
#linear-inequalities
#interval-form
#negative-division
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A \([-2,\infty\))
B (\(-\infty,-2]\)
C (\(-2,\infty\))
D (\(-\infty,-2\))
Explanation opens after your attempt
Correct Answer
A. \([-2,\infty\))
Step 1
Concept
Solving gives \(-3x\le 6\) and then \(x\ge -2\). Remember to reverse the sign when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is A. \([-2,\infty\)). Solving gives \(-3x\le 6\) and then \(x\ge -2\). Remember to reverse the sign when dividing by a negative.
Step 3
Exam Tip
हल करने पर \(-3x\le 6\) और फिर \(x\ge -2\) मिलता है। ऋणात्मक से भाग देते समय चिह्न बदलना न भूलें।
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असमानता \(5x-1\le 14\) का अंतराल रूप कौन सा है?
Which is the interval form of \(5x-1\le 14\)?
#linear-inequalities
#interval-notation
#less-than-equal
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A (\(-\infty,3\))
B (\(-\infty,3]\)
C \([3,\infty\))
D (\(3,\infty\))
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,3]\)
Step 1
Concept
Solving gives \(5x\le 15\) and \(x\le 3\). The sign \(\le\) includes (3).
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,3]\). Solving gives \(5x\le 15\) and \(x\le 3\). The sign \(\le\) includes (3).
Step 3
Exam Tip
हल करने पर \(5x\le 15\) और \(x\le 3\) मिलता है। \(\le\) में (3) शामिल होगा।
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यदि संख्या रेखा पर (-1) और (6) के बीच छायांकन है तथा (-1) खुला और (6) बंद है, तो असमता कौन सी है?
If the number line is shaded between (-1) and (6), with (-1) open and (6) closed, which inequality is correct?
#linear-inequalities
#graph-to-inequality
#mixed-endpoints
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A \(-1\le x<6\)
B (-1<x<6)
C \(-1<x\le 6\)
D \(-1\le x\le 6\)
Explanation opens after your attempt
Correct Answer
C. \(-1<x\le 6\)
Step 1
Concept
An open endpoint excludes the boundary and a closed endpoint includes it. Hence \(-1<x\le 6\) is correct.
Step 2
Why this answer is correct
The correct answer is C. \(-1<x\le 6\). An open endpoint excludes the boundary and a closed endpoint includes it. Hence \(-1<x\le 6\) is correct.
Step 3
Exam Tip
खुला सिरा सीमा को बाहर रखता है और बंद सिरा सीमा को शामिल करता है। इसलिए \(-1<x\le 6\) सही है।
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संख्या रेखा पर ([ -4,3 )) का सही असमता रूप कौन सा है?
Which inequality form correctly represents ([-4,3)) on the number line?
#linear-inequalities
#interval-to-inequality
#number-line
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A (-4<x<3)
B \(-4\le x<3\)
C \(-4<x\le 3\)
D \(-4\le x\le 3\)
Explanation opens after your attempt
Correct Answer
B. \(-4\le x<3\)
Step 1
Concept
The square bracket includes (-4) and the parenthesis excludes (3). Therefore the form is \(-4\le x<3\).
Step 2
Why this answer is correct
The correct answer is B. \(-4\le x<3\). The square bracket includes (-4) and the parenthesis excludes (3). Therefore the form is \(-4\le x<3\).
Step 3
Exam Tip
वर्ग कोष्ठक (-4) को शामिल करता है और गोल कोष्ठक (3) को बाहर रखता है। इसलिए \(-4\le x<3\) होगा।
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\(-6\le 2x<10\) को हल करके संख्या रेखा पर कौन सा अंतराल मिलेगा?
After solving \(-6\le 2x<10\), which interval is obtained on the number line?
#linear-inequalities
#compound-inequality
#interval-form
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A ((-3,5))
B ([-3,5))
C ((-3,5])
D ([-3,5])
Explanation opens after your attempt
Correct Answer
B. ([-3,5))
Step 1
Concept
Dividing all parts by (2) gives \(-3\le x<5\). Hence the left endpoint is closed and the right endpoint is open.
Step 2
Why this answer is correct
The correct answer is B. ([-3,5)). Dividing all parts by (2) gives \(-3\le x<5\). Hence the left endpoint is closed and the right endpoint is open.
Step 3
Exam Tip
सभी भागों को (2) से भाग देने पर \(-3\le x<5\) मिलता है। इसलिए बायाँ सिरा बंद और दायाँ सिरा खुला है।
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संख्या रेखा पर \(x\le -2\) या (x>4) का अंतराल रूप कौन सा है?
Which interval form represents \(x\le -2\) or (x>4) on the number line?
#linear-inequalities
#union
#interval-notation
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A ((-\infty,-2]\cup\(4,\infty\))
B (\(-\infty,-2\)\cup[4,\infty))
C ([-2,4])
D ((-2,4))
Explanation opens after your attempt
Correct Answer
A. ((-\infty,-2]\cup\(4,\infty\))
Step 1
Concept
The first part includes (-2) and the second part excludes (4). Use \(\cup\) for separate parts.
Step 2
Why this answer is correct
The correct answer is A. ((-\infty,-2]\cup\(4,\infty\)). The first part includes (-2) and the second part excludes (4). Use \(\cup\) for separate parts.
Step 3
Exam Tip
पहला भाग (-2) को शामिल करता है और दूसरा भाग (4) को बाहर रखता है। अलग भागों के लिए \(\cup\) का प्रयोग करें।
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असमानता \(\frac{x-2}{3}\ge 1\) का संख्या रेखा निरूपण कौन सा होगा?
Which number line representation corresponds to \(\frac{x-2}{3}\ge 1\)?
#linear-inequalities
#fraction
#number-line
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A (5) पर खाली वृत्त और दाईं ओर / Open circle at (5) and right side
B (5) पर भरा वृत्त और दाईं ओर / Closed circle at (5) and right side
C (5) पर भरा वृत्त और बाईं ओर / Closed circle at (5) and left side
D (5) पर खाली वृत्त और बाईं ओर / Open circle at (5) and left side
Explanation opens after your attempt
Correct Answer
B. (5) पर भरा वृत्त और दाईं ओर / Closed circle at (5) and right side
Step 1
Concept
Solving gives \(x-2\ge 3\) and \(x\ge 5\). Since the boundary is included, use a closed circle.
Step 2
Why this answer is correct
The correct answer is B. (5) पर भरा वृत्त और दाईं ओर / Closed circle at (5) and right side. Solving gives \(x-2\ge 3\) and \(x\ge 5\). Since the boundary is included, use a closed circle.
Step 3
Exam Tip
हल करने पर \(x-2\ge 3\) और \(x\ge 5\) मिलता है। सीमा शामिल होने से भरा वृत्त बनेगा।
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असमानता \(\frac{2x+1}{5}<3\) का सही अंतराल कौन सा है?
Which is the correct interval for \(\frac{2x+1}{5}<3\)?
#linear-inequalities
#fraction
#interval-form
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A (\(-\infty,7\))
B (\(-\infty,7]\)
C (\(7,\infty\))
D \([7,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,7\))
Step 1
Concept
Solving gives (2x+1<15) and (x<7). A strict inequality does not include (7).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,7\)). Solving gives (2x+1<15) and (x<7). A strict inequality does not include (7).
Step 3
Exam Tip
हल करने पर (2x+1<15) और (x<7) मिलता है। कठोर असमता में (7) शामिल नहीं होगा।
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यदि संख्या रेखा पर (-5) पर खाली वृत्त और (1) पर खाली वृत्त है तथा बाहर की ओर छायांकन है, तो कौन सा अंतराल सही है?
If the number line has open circles at (-5) and (1) with shading outward, which interval is correct?
#linear-inequalities
#outside-region
#union
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A ((-5,1))
B (\(-\infty,-5\)\cup\(1,\infty\))
C (\(-\infty,-5]\cup[1,\infty\))
D ([-5,1])
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,-5\)\cup\(1,\infty\))
Step 1
Concept
Outward shading shows two separate regions and open circles exclude the endpoints. Hence (\(-\infty,-5\)\cup\(1,\infty\)) is correct.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,-5\)\cup\(1,\infty\)). Outward shading shows two separate regions and open circles exclude the endpoints. Hence (\(-\infty,-5\)\cup\(1,\infty\)) is correct.
Step 3
Exam Tip
बाहर की छायांकन दो अलग क्षेत्रों को दिखाती है और खुले वृत्त सीमाओं को बाहर रखते हैं। इसलिए (\(-\infty,-5\)\cup\(1,\infty\)) सही है।
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यदि संख्या रेखा पर (-7) और (3) पर भरे वृत्त हैं तथा बाहर की ओर छायांकन है, तो निरूपण कौन सा है?
If the number line has closed circles at (-7) and (3) with outward shading, which representation is correct?
#linear-inequalities
#outside-interval
#closed-endpoints
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A (x<-7) या (x>3) / (x<-7) or (x>3)
B \(x\le -7\) या \(x\ge 3\) / \(x\le -7\) or \(x\ge 3\)
C (-7<x<3)
D \(-7\le x\le 3\)
Explanation opens after your attempt
Correct Answer
B. \(x\le -7\) या \(x\ge 3\) / \(x\le -7\) or \(x\ge 3\)
Step 1
Concept
Closed circles include both endpoints and outward shading shows two outside regions. So inclusive inequalities with or are used.
Step 2
Why this answer is correct
The correct answer is B. \(x\le -7\) या \(x\ge 3\) / \(x\le -7\) or \(x\ge 3\). Closed circles include both endpoints and outward shading shows two outside regions. So inclusive inequalities with or are used.
Step 3
Exam Tip
भरे वृत्त दोनों सीमाओं को शामिल करते हैं और बाहर की छायांकन दो बाहरी क्षेत्रों को दिखाती है। इसलिए बराबरी सहित या का प्रयोग होगा।
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\(-2\le \frac{x-1}{3}<4\) का संख्या रेखा अंतराल क्या है?
What is the number line interval for \(-2\le \frac{x-1}{3}<4\)?
#linear-inequalities
#compound
#fraction
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A ([-5,13))
B ((-5,13])
C ([-7,11))
D ((-7,11])
Explanation opens after your attempt
Correct Answer
A. ([-5,13))
Step 1
Concept
First multiply by (3) and then add (1) to get \(-5\le x<13\). Therefore the interval is ([-5,13)).
Step 2
Why this answer is correct
The correct answer is A. ([-5,13)). First multiply by (3) and then add (1) to get \(-5\le x<13\). Therefore the interval is ([-5,13)).
Step 3
Exam Tip
पहले (3) से गुणा करें और फिर (1) जोड़ें तो \(-5\le x<13\) मिलता है। इसलिए अंतराल ([-5,13)) है।
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संख्या रेखा पर (x\in(-4,8]) और \(x\in[0,10\)) का समान भाग कौन सा है?
What is the common part of (x\in(-4,8]) and \(x\in[0,10\)) on the number line?
#linear-inequalities
#intersection
#intervals
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A ((-4,10))
B ([0,8])
C ((0,8))
D ([-4,8])
Explanation opens after your attempt
Correct Answer
B. ([0,8])
Step 1
Concept
The common part of both intervals is from (0) to (8), and both values are included. Hence ([0,8]) is correct.
Step 2
Why this answer is correct
The correct answer is B. ([0,8]). The common part of both intervals is from (0) to (8), and both values are included. Hence ([0,8]) is correct.
Step 3
Exam Tip
दोनों अंतरालों में सामान्य भाग (0) से (8) तक है और दोनों मान शामिल हैं। इसलिए ([0,8]) सही है।
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संख्या रेखा पर \(x\in[-6,2\)) और (x\in(2,5]) का समान भाग क्या है?
What is the common part of \(x\in[-6,2\)) and (x\in(2,5]) on the number line?
#linear-inequalities
#empty-intersection
#number-line
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A ({2})
B ([-6,5])
C \(\emptyset\)
D ((2,5])
Explanation opens after your attempt
Correct Answer
C. \(\emptyset\)
Step 1
Concept
The first interval ends before (2), and the second starts after (2). Therefore there is no common point.
Step 2
Why this answer is correct
The correct answer is C. \(\emptyset\). The first interval ends before (2), and the second starts after (2). Therefore there is no common point.
Step 3
Exam Tip
पहला अंतराल (2) से पहले खत्म होता है और दूसरा (2) के बाद शुरू होता है। इसलिए कोई सामान्य बिंदु नहीं है।
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यदि (x) पूर्णांक है और \(1\le x<7\), तो संख्या रेखा पर कौन से बिंदु चिह्नित होंगे?
If (x) is an integer and \(1\le x<7\), which points will be marked on the number line?
#linear-inequalities
#integer-solutions
#number-line
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A (1,2,3,4,5,6)
B (2,3,4,5,6,7)
C (1,2,3,4,5,6,7)
D (2,3,4,5,6)
Explanation opens after your attempt
Correct Answer
A. (1,2,3,4,5,6)
Step 1
Concept
(1) is included but (7) is not. So the integers are from (1) to (6).
Step 2
Why this answer is correct
The correct answer is A. (1,2,3,4,5,6). (1) is included but (7) is not. So the integers are from (1) to (6).
Step 3
Exam Tip
(1) शामिल है लेकिन (7) शामिल नहीं है। इसलिए पूर्णांक (1) से (6) तक होंगे।
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असमानता (7-2x>1) के लिए संख्या रेखा पर सही अंतराल कौन सा है?
Which interval is correct on the number line for (7-2x>1)?
#linear-inequalities
#negative-division
#interval
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A (\(-\infty,3\))
B (\(-\infty,3]\)
C (\(3,\infty\))
D \([3,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,3\))
Step 1
Concept
Solving gives (-2x>-6) and then (x<3). The sign reverses when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,3\)). Solving gives (-2x>-6) and then (x<3). The sign reverses when dividing by a negative.
Step 3
Exam Tip
हल करने पर (-2x>-6) और फिर (x<3) मिलता है। ऋणात्मक से भाग देने पर चिह्न उलटता है।
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असमानता \(6-4x\ge -10\) को संख्या रेखा पर कौन सा अंतराल दिखाएगा?
Which interval shows \(6-4x\ge -10\) on the number line?
#linear-inequalities
#negative-coefficient
#interval-form
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A (\(-\infty,4\))
B (\(-\infty,4]\)
C \([4,\infty\))
D (\(4,\infty\))
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,4]\)
Step 1
Concept
Solving gives \(-4x\ge -16\) and \(x\le 4\). Since equality is included, (4) is a closed endpoint.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,4]\). Solving gives \(-4x\ge -16\) and \(x\le 4\). Since equality is included, (4) is a closed endpoint.
Step 3
Exam Tip
हल करने पर \(-4x\ge -16\) और \(x\le 4\) मिलता है। बराबरी शामिल होने से (4) पर बंद सिरा होगा।
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\(2<3x-1\le 11\) के संख्या रेखा निरूपण का सही अंतराल कौन सा है?
Which interval is correct for the number line representation of \(2<3x-1\le 11\)?
#linear-inequalities
#compound
#interval
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A ((1,4])
B ([1,4))
C ((2,11])
D ([2,11))
Explanation opens after your attempt
Correct Answer
A. ((1,4])
Step 1
Concept
Adding (1) gives \(3<3x\le 12\), then \(1<x\le 4\). Hence ((1,4]) is correct.
Step 2
Why this answer is correct
The correct answer is A. ((1,4]). Adding (1) gives \(3<3x\le 12\), then \(1<x\le 4\). Hence ((1,4]) is correct.
Step 3
Exam Tip
(1) जोड़कर \(3<3x\le 12\) और फिर \(1<x\le 4\) मिलता है। इसलिए ((1,4]) सही है।
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संख्या रेखा पर (x\in\(-\infty,0]\cap[-3,4\)) का परिणाम क्या है?
What is the result of (x\in\(-\infty,0]\cap[-3,4\)) on the number line?
#linear-inequalities
#intersection
#interval-operation
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A ([-3,0])
B ((-3,0))
C ([0,4))
D (\(-\infty,4\))
Explanation opens after your attempt
Correct Answer
A. ([-3,0])
Step 1
Concept
The common part is from (-3) to (0). Both endpoints are included in this intersection.
Step 2
Why this answer is correct
The correct answer is A. ([-3,0]). The common part is from (-3) to (0). Both endpoints are included in this intersection.
Step 3
Exam Tip
दोनों का समान भाग (-3) से (0) तक है। दोनों सीमाएँ इस समान भाग में शामिल हैं।
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यदि (x\in\(1,5]\cup[5,9\)), तो संख्या रेखा पर सरल अंतराल कौन सा है?
If (x\in\(1,5]\cup[5,9\)), what is the simplified interval on the number line?
#linear-inequalities
#union
#simplification
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A ((1,9))
B ([1,9))
C ((1,9])
D ([1,9])
Explanation opens after your attempt
Correct Answer
A. ((1,9))
Step 1
Concept
The intervals meet at (5), while (1) and (9) are excluded. Together they form ((1,9)).
Step 2
Why this answer is correct
The correct answer is A. ((1,9)). The intervals meet at (5), while (1) and (9) are excluded. Together they form ((1,9)).
Step 3
Exam Tip
दोनों अंतराल (5) पर जुड़े हुए हैं और (1) तथा (9) शामिल नहीं हैं। इसलिए मिलकर ((1,9)) बनता है।
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असमानता \(-1\le \frac{x+2}{4}\le 3\) का सही संख्या रेखा अंतराल क्या है?
What is the correct number line interval for \(-1\le \frac{x+2}{4}\le 3\)?
#linear-inequalities
#compound
#fraction
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A ([-6,10])
B ((-6,10))
C ([-2,12])
D ((-2,12))
Explanation opens after your attempt
Correct Answer
A. ([-6,10])
Step 1
Concept
Multiplying by (4) gives \(-4\le x+2\le 12\), then \(-6\le x\le 10\). Both endpoints are included.
Step 2
Why this answer is correct
The correct answer is A. ([-6,10]). Multiplying by (4) gives \(-4\le x+2\le 12\), then \(-6\le x\le 10\). Both endpoints are included.
Step 3
Exam Tip
(4) से गुणा करने पर \(-4\le x+2\le 12\) और फिर \(-6\le x\le 10\) मिलता है। दोनों सीमाएँ शामिल हैं।
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संख्या रेखा पर \(x\in[-2,6]\setminus{3}\) का सही वर्णन कौन सा है?
Which description correctly represents \(x\in[-2,6]\setminus{3}\) on the number line?
#linear-inequalities
#set-difference
#number-line
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A (-2) से (6) तक छायांकन और (3) पर खाली वृत्त / Shading from (-2) to (6) with an open circle at (3)
B (-2) से (6) तक छायांकन और सभी बिंदु भरे / Shading from (-2) to (6) with all points closed
C केवल (3) पर भरा बिंदु / Only a closed point at (3)
D पूरी संख्या रेखा छायांकित / Entire number line shaded
Explanation opens after your attempt
Correct Answer
A. (-2) से (6) तक छायांकन और (3) पर खाली वृत्त / Shading from (-2) to (6) with an open circle at (3)
Step 1
Concept
The notation \(\setminus{3}\) means remove (3). So an open circle appears at (3) inside the closed interval.
Step 2
Why this answer is correct
The correct answer is A. (-2) से (6) तक छायांकन और (3) पर खाली वृत्त / Shading from (-2) to (6) with an open circle at (3). The notation \(\setminus{3}\) means remove (3). So an open circle appears at (3) inside the closed interval.
Step 3
Exam Tip
\(\setminus{3}\) का अर्थ है (3) को हटाना। इसलिए बंद अंतराल में (3) पर खाली वृत्त दिखेगा।
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यदि (x\in\(-\infty,2\)\cup\(2,\infty\)), तो कौन सा कथन सही है?
If (x\in\(-\infty,2\)\cup\(2,\infty\)), which statement is correct?
#linear-inequalities
#excluded-point
#union
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A (x=2) शामिल है / (x=2) is included
B (x=2) शामिल नहीं है / (x=2) is not included
C केवल (x=2) समाधान है / Only (x=2) is the solution
D कोई वास्तविक संख्या शामिल नहीं है / No real number is included
Explanation opens after your attempt
Correct Answer
B. (x=2) शामिल नहीं है / (x=2) is not included
Step 1
Concept
It takes all numbers on both sides but leaves out (2). On the number line, (2) has an open circle.
Step 2
Why this answer is correct
The correct answer is B. (x=2) शामिल नहीं है / (x=2) is not included. It takes all numbers on both sides but leaves out (2). On the number line, (2) has an open circle.
Step 3
Exam Tip
यह दोनों ओर की सभी संख्याएँ लेता है लेकिन (2) को छोड़ देता है। संख्या रेखा पर (2) पर खाली वृत्त होगा।
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(|x|<3) का संख्या रेखा पर सही अंतराल कौन सा है?
Which interval correctly represents (|x|<3) on the number line?
#linear-inequalities
#absolute-value
#number-line
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A ((-3,3))
B ([-3,3])
C (\(-\infty,-3\)\cup\(3,\infty\))
D (\(-\infty,3\))
Explanation opens after your attempt
Correct Answer
A. ((-3,3))
Step 1
Concept
(|x|<3) means (x) lies between (-3) and (3). Because the inequality is strict, both endpoints are open.
Step 2
Why this answer is correct
The correct answer is A. ((-3,3)). (|x|<3) means (x) lies between (-3) and (3). Because the inequality is strict, both endpoints are open.
Step 3
Exam Tip
(|x|<3) का अर्थ है (x), (-3) और (3) के बीच है। कठोर असमता के कारण दोनों सिरे खुले होंगे।
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\(|x|\le 5\) को संख्या रेखा पर किस अंतराल से दिखाएँगे?
Which interval shows \(|x|\le 5\) on the number line?
#linear-inequalities
#absolute-value
#closed-interval
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A ((-5,5))
B ([-5,5])
C (\(-\infty,-5]\cup[5,\infty\))
D \([5,\infty\))
Explanation opens after your attempt
Correct Answer
B. ([-5,5])
Step 1
Concept
\(|x|\le 5\) means \(-5\le x\le 5\). Since equality is present, both endpoints are closed.
Step 2
Why this answer is correct
The correct answer is B. ([-5,5]). \(|x|\le 5\) means \(-5\le x\le 5\). Since equality is present, both endpoints are closed.
Step 3
Exam Tip
\(|x|\le 5\) का अर्थ है \(-5\le x\le 5\)। बराबरी होने से दोनों सिरे बंद होंगे।
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(|x|>4) का संख्या रेखा पर सही निरूपण कौन सा है?
Which is the correct number line representation of (|x|>4)?
#linear-inequalities
#absolute-value
#outside-region
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A ((-4,4))
B ([-4,4])
C (\(-\infty,-4\)\cup\(4,\infty\))
D (\(-\infty,4\))
Explanation opens after your attempt
Correct Answer
C. (\(-\infty,-4\)\cup\(4,\infty\))
Step 1
Concept
(|x|>4) means (x<-4) or (x>4). Both boundaries are excluded, so open circles are used.
Step 2
Why this answer is correct
The correct answer is C. (\(-\infty,-4\)\cup\(4,\infty\)). (|x|>4) means (x<-4) or (x>4). Both boundaries are excluded, so open circles are used.
Step 3
Exam Tip
(|x|>4) में (x<-4) या (x>4) होता है। दोनों सीमाएँ शामिल नहीं हैं इसलिए खुले वृत्त होंगे।
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\(|x|\ge 2\) के लिए संख्या रेखा पर कौन सा अंतराल सही है?
Which interval is correct on the number line for \(|x|\ge 2\)?
#linear-inequalities
#absolute-value
#union
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A ((-2,2))
B ([-2,2])
C (\(-\infty,-2]\cup[2,\infty\))
D (\(-\infty,-2\)\cup\(2,\infty\))
Explanation opens after your attempt
Correct Answer
C. (\(-\infty,-2]\cup[2,\infty\))
Step 1
Concept
\(|x|\ge 2\) means \(x\le -2\) or \(x\ge 2\). Equality is included, so the outer endpoints are closed.
Step 2
Why this answer is correct
The correct answer is C. (\(-\infty,-2]\cup[2,\infty\)). \(|x|\ge 2\) means \(x\le -2\) or \(x\ge 2\). Equality is included, so the outer endpoints are closed.
Step 3
Exam Tip
\(|x|\ge 2\) में \(x\le -2\) या \(x\ge 2\) होता है। बराबरी शामिल है इसलिए दोनों बाहरी सिरे बंद होंगे।
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संख्या रेखा पर \(x\in[-8,-2]\cup(-2,5)\) का सरल रूप कौन सा है?
What is the simplified form of \(x\in[-8,-2]\cup(-2,5)\) on the number line?
#linear-inequalities
#union-simplification
#interval
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A ([-8,5))
B ((-8,5))
C ([-8,5])
D ((-8,5])
Explanation opens after your attempt
Correct Answer
A. ([-8,5))
Step 1
Concept
The first interval includes (-2), and the second starts just after it. Together they form ([-8,5)).
Step 2
Why this answer is correct
The correct answer is A. ([-8,5)). The first interval includes (-2), and the second starts just after it. Together they form ([-8,5)).
Step 3
Exam Tip
पहले अंतराल में (-2) शामिल है और दूसरा उसके तुरंत बाद शुरू होता है। मिलकर ([-8,5)) बनता है।
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संख्या रेखा पर (x\in(-\infty,3]\cap\(0,\infty\)) का सही अंतराल क्या है?
What is the correct interval for (x\in(-\infty,3]\cap\(0,\infty\)) on the number line?
#linear-inequalities
#intersection
#number-line
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A ((0,3])
B ([0,3])
C ((0,3))
D ([0,3))
Explanation opens after your attempt
Correct Answer
A. ((0,3])
Step 1
Concept
The common part is greater than (0) and up to (3). (0) is excluded but (3) is included.
Step 2
Why this answer is correct
The correct answer is A. ((0,3]). The common part is greater than (0) and up to (3). (0) is excluded but (3) is included.
Step 3
Exam Tip
समान भाग (0) से बड़ा और (3) तक है। (0) शामिल नहीं है लेकिन (3) शामिल है।
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असमानता (2(x-1)\le x+4) का संख्या रेखा अंतराल कौन सा है?
Which number line interval represents (2(x-1)\le x+4)?
#linear-inequalities
#linear-expression
#interval
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A (\(-\infty,6]\)
B \([6,\infty\))
C (\(-\infty,6\))
D (\(6,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,6]\)
Step 1
Concept
Solving gives \(2x-2\le x+4\) and \(x\le 6\). Since \(\le\) is present, (6) is included.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,6]\). Solving gives \(2x-2\le x+4\) and \(x\le 6\). Since \(\le\) is present, (6) is included.
Step 3
Exam Tip
हल करने पर \(2x-2\le x+4\) और \(x\le 6\) मिलता है। \(\le\) होने से (6) शामिल है।
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असमानता (3(x+2)>2x-1) का संख्या रेखा पर सही रूप कौन सा है?
Which is the correct number line form for (3(x+2)>2x-1)?
#linear-inequalities
#linear-expression
#number-line
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A (\(-\infty,-7\))
B (\(-7,\infty\))
C \([-7,\infty\))
D (\(-\infty,-7]\)
Explanation opens after your attempt
Correct Answer
B. (\(-7,\infty\))
Step 1
Concept
Solving gives (3x+6>2x-1) and (x>-7). Since the inequality is strict, (-7) remains open.
Step 2
Why this answer is correct
The correct answer is B. (\(-7,\infty\)). Solving gives (3x+6>2x-1) and (x>-7). Since the inequality is strict, (-7) remains open.
Step 3
Exam Tip
हल करने पर (3x+6>2x-1) और (x>-7) मिलता है। कठोर असमता होने से (-7) खुला रहेगा।
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असमानता (5-2(x+1)\ge 1) को संख्या रेखा पर किस अंतराल से दिखाएँगे?
Which interval shows (5-2(x+1)\ge 1) on the number line?
#linear-inequalities
#negative-division
#interval
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A (\(-\infty,1]\)
B \([1,\infty\))
C (\(-\infty,1\))
D (\(1,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,1]\)
Step 1
Concept
Simplifying gives \(3-2x\ge 1\), then \(-2x\ge -2\), and \(x\le 1\). The sign reverses when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,1]\). Simplifying gives \(3-2x\ge 1\), then \(-2x\ge -2\), and \(x\le 1\). The sign reverses when dividing by a negative.
Step 3
Exam Tip
सरल करने पर \(3-2x\ge 1\), फिर \(-2x\ge -2\) और \(x\le 1\) मिलता है। ऋणात्मक से भाग देने पर चिह्न बदलता है।
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यदि संख्या रेखा पर (2) पर खाली वृत्त और (9) पर खाली वृत्त है तथा बीच में छायांकन है, तो कौन सा कथन सही है?
If the number line has open circles at (2) and (9) with shading between them, which statement is correct?
#linear-inequalities
#open-endpoints
#number-line
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A (2) और (9) दोनों शामिल हैं / Both (2) and (9) are included
B (2) शामिल है लेकिन (9) नहीं / (2) is included but (9) is not
C (2) और (9) दोनों शामिल नहीं हैं / Both (2) and (9) are not included
D केवल (9) शामिल है / Only (9) is included
Explanation opens after your attempt
Correct Answer
C. (2) और (9) दोनों शामिल नहीं हैं / Both (2) and (9) are not included
Step 1
Concept
Open circles keep boundary points out of the solution. Therefore both endpoints are not included.
Step 2
Why this answer is correct
The correct answer is C. (2) और (9) दोनों शामिल नहीं हैं / Both (2) and (9) are not included. Open circles keep boundary points out of the solution. Therefore both endpoints are not included.
Step 3
Exam Tip
खाली वृत्त सीमा को समाधान से बाहर रखता है। इसलिए दोनों अंतिम बिंदु शामिल नहीं होंगे।
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अंतराल ([-3,7]) से (0) हटाने पर संख्या रेखा पर कौन सा रूप मिलेगा?
After removing (0) from the interval ([-3,7]), which form appears on the number line?
#linear-inequalities
#set-difference
#interval
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A ([-3,0)\cup(0,7])
B \([-3,0]\cup[0,7]\)
C \((-3,0)\cup(0,7)\)
D ([-3,7])
Explanation opens after your attempt
Correct Answer
A. ([-3,0)\cup(0,7])
Step 1
Concept
Removing (0) means placing an open circle at (0). The endpoints (-3) and (7) remain included.
Step 2
Why this answer is correct
The correct answer is A. ([-3,0)\cup(0,7]). Removing (0) means placing an open circle at (0). The endpoints (-3) and (7) remain included.
Step 3
Exam Tip
(0) हटाने का अर्थ है (0) पर खाली वृत्त लगाना। बाकी (-3) और (7) शामिल रहेंगे।
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यदि \(x\in[-10,-4\)\cup[-4,2]), तो संयुक्त संख्या रेखा अंतराल कौन सा है?
If \(x\in[-10,-4\)\cup[-4,2]), what is the combined number line interval?
#linear-inequalities
#union-simplification
#closed-interval
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A ([-10,2])
B ([-10,2))
C ((-10,2])
D ((-10,2))
Explanation opens after your attempt
Correct Answer
A. ([-10,2])
Step 1
Concept
The second interval includes (-4), so no gap remains in the middle. The combined interval is ([-10,2]).
Step 2
Why this answer is correct
The correct answer is A. ([-10,2]). The second interval includes (-4), so no gap remains in the middle. The combined interval is ([-10,2]).
Step 3
Exam Tip
दूसरा अंतराल (-4) को शामिल करता है इसलिए बीच में कोई खाली जगह नहीं बचती। संयुक्त अंतराल ([-10,2]) है।
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संख्या रेखा पर \(x\in(-2,6)\cap[1,8]\) का परिणाम क्या होगा?
What is the result of \(x\in(-2,6)\cap[1,8]\) on the number line?
#linear-inequalities
#intersection
#mixed-endpoints
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A ([1,6))
B ((1,6])
C ((1,6))
D ([-2,8])
Explanation opens after your attempt
Correct Answer
A. ([1,6))
Step 1
Concept
The common part starts at (1) and goes up to but not including (6). (1) is included and (6) is excluded.
Step 2
Why this answer is correct
The correct answer is A. ([1,6)). The common part starts at (1) and goes up to but not including (6). (1) is included and (6) is excluded.
Step 3
Exam Tip
समान भाग (1) से शुरू होकर (6) से पहले तक है। (1) शामिल है और (6) शामिल नहीं है।
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यदि (x) वास्तविक संख्या है और \(x\notin[2,5]\), तो संख्या रेखा पर सही अंतराल कौन सा है?
If (x) is a real number and \(x\notin[2,5]\), which interval is correct on the number line?
#linear-inequalities
#complement
#number-line
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A (\(-\infty,2\)\cup\(5,\infty\))
B (\(-\infty,2]\cup[5,\infty\))
C ([2,5])
D ((2,5))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,2\)\cup\(5,\infty\))
Step 1
Concept
Since ([2,5]) includes (2) and (5), the outside part will not include them. Thus both boundaries are open.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,2\)\cup\(5,\infty\)). Since ([2,5]) includes (2) and (5), the outside part will not include them. Thus both boundaries are open.
Step 3
Exam Tip
([2,5]) में (2) और (5) शामिल हैं इसलिए बाहर का भाग उन्हें शामिल नहीं करेगा। इसलिए दोनों सीमाएँ खुली रहेंगी।
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यदि (x) वास्तविक संख्या है और \(x\notin(-1,4)\), तो सही संख्या रेखा निरूपण क्या है?
If (x) is a real number and \(x\notin(-1,4)\), what is the correct number line representation?
#linear-inequalities
#complement
#closed-endpoints
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A (\(-\infty,-1\)\cup\(4,\infty\))
B (\(-\infty,-1]\cup[4,\infty\))
C ((-1,4))
D ([-1,4])
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,-1]\cup[4,\infty\))
Step 1
Concept
((-1,4)) does not include (-1) and (4), so the complement includes both. Hence the outside regions have closed endpoints.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,-1]\cup[4,\infty\)). ((-1,4)) does not include (-1) and (4), so the complement includes both. Hence the outside regions have closed endpoints.
Step 3
Exam Tip
((-1,4)) में (-1) और (4) शामिल नहीं हैं इसलिए पूरक में दोनों शामिल होंगे। अतः बाहरी भाग बंद सिरों के साथ होगा।
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असमानता \(\frac{5-2x}{3}\le 1\) का संख्या रेखा निरूपण कौन सा होगा?
Which number line representation corresponds to \(\frac{5-2x}{3}\le 1\)?
#linear-inequalities
#fraction
#negative-division
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A (1) पर खाली वृत्त और बाईं ओर / Open circle at (1) and left side
B (1) पर भरा वृत्त और दाईं ओर / Closed circle at (1) and right side
C (1) पर भरा वृत्त और बाईं ओर / Closed circle at (1) and left side
D (1) पर खाली वृत्त और दाईं ओर / Open circle at (1) and right side
Explanation opens after your attempt
Correct Answer
B. (1) पर भरा वृत्त और दाईं ओर / Closed circle at (1) and right side
Step 1
Concept
Solving gives \(5-2x\le 3\), then \(-2x\le -2\), and \(x\ge 1\). The boundary is included, so (1) is closed.
Step 2
Why this answer is correct
The correct answer is B. (1) पर भरा वृत्त और दाईं ओर / Closed circle at (1) and right side. Solving gives \(5-2x\le 3\), then \(-2x\le -2\), and \(x\ge 1\). The boundary is included, so (1) is closed.
Step 3
Exam Tip
हल करने पर \(5-2x\le 3\), फिर \(-2x\le -2\) और \(x\ge 1\) मिलता है। सीमा शामिल है इसलिए (1) बंद होगा।
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संख्या रेखा पर (x\in[-5,0]\cup(0,4]) का सरल रूप कौन सा है?
What is the simplified form of (x\in[-5,0]\cup(0,4]) on the number line?
#linear-inequalities
#union-simplification
#number-line
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A ([-5,4])
B ([-5,4))
C ((-5,4])
D ((-5,4))
Explanation opens after your attempt
Correct Answer
A. ([-5,4])
Step 1
Concept
The first interval includes (0), and the second starts after (0). So no gap remains and the result is ([-5,4]).
Step 2
Why this answer is correct
The correct answer is A. ([-5,4]). The first interval includes (0), and the second starts after (0). So no gap remains and the result is ([-5,4]).
Step 3
Exam Tip
पहले अंतराल में (0) शामिल है और दूसरा (0) के बाद शुरू होता है। इसलिए कोई खाली जगह नहीं रहती और ([-5,4]) बनता है।
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यदि \(-2\le x<5\) और (x) प्राकृतिक संख्या है, तो संख्या रेखा पर कितने प्राकृतिक बिंदु आएँगे?
If \(-2\le x<5\) and (x) is a natural number, how many natural number points appear on the number line?
#linear-inequalities
#natural-numbers
#number-line
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A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The natural numbers (1,2,3,4) lie in this interval. (5) is not included.
Step 2
Why this answer is correct
The correct answer is B. (4). The natural numbers (1,2,3,4) lie in this interval. (5) is not included.
Step 3
Exam Tip
प्राकृतिक संख्याएँ (1,2,3,4) इस अंतराल में आती हैं। (5) शामिल नहीं है।
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अंतराल (\(-\infty,1]\cap[1,\infty\)) संख्या रेखा पर क्या दर्शाता है?
What does the interval (\(-\infty,1]\cap[1,\infty\)) represent on the number line?
#linear-inequalities
#intersection
#singleton
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A ({1})
B (\(-\infty,\infty\))
C \(\emptyset\)
D (\(1,\infty\))
Explanation opens after your attempt
Step 1
Concept
The only common point of the two intervals is (1). So the number line has a closed point at (1).
Step 2
Why this answer is correct
The correct answer is A. ({1}). The only common point of the two intervals is (1). So the number line has a closed point at (1).
Step 3
Exam Tip
दोनों अंतरालों का केवल सामान्य बिंदु (1) है। इसलिए संख्या रेखा पर (1) पर भरा बिंदु होगा।
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यदि संख्या रेखा पर (a) पर खुला सिरा और (b) पर बंद सिरा है तथा (a<b), तो कौन सा अंतराल सही है?
If a number line has an open endpoint at (a) and a closed endpoint at (b), where (a<b), which interval is correct?
#linear-inequalities
#general-interval
#number-line
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A ([a,b])
B ((a,b])
C ([a,b))
D ((a,b))
Explanation opens after your attempt
Correct Answer
B. ((a,b])
Step 1
Concept
The open endpoint excludes (a), and the closed endpoint includes (b). Therefore the interval is ((a,b]).
Step 2
Why this answer is correct
The correct answer is B. ((a,b]). The open endpoint excludes (a), and the closed endpoint includes (b). Therefore the interval is ((a,b]).
Step 3
Exam Tip
खुला सिरा (a) को बाहर रखता है और बंद सिरा (b) को शामिल करता है। इसलिए अंतराल ((a,b]) होगा।
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असमानता (4x+1>9) को हल करने पर संख्या रेखा पर कौन सा निरूपण सही होगा?
After solving the inequality (4x+1>9), which representation will be correct on the number line?
#linear-inequalities
#solving
#number-line
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A (2) पर खाली वृत्त और दाईं ओर छायांकन / Open circle at (2) and shading right
B (2) पर भरा वृत्त और दाईं ओर छायांकन / Closed circle at (2) and shading right
C (2) पर खाली वृत्त और बाईं ओर छायांकन / Open circle at (2) and shading left
D (2) पर भरा वृत्त और बाईं ओर छायांकन / Closed circle at (2) and shading left
Explanation opens after your attempt
Correct Answer
A. (2) पर खाली वृत्त और दाईं ओर छायांकन / Open circle at (2) and shading right
Step 1
Concept
Solving gives (x>2). A strict inequality excludes the boundary so an open circle is used.
Step 2
Why this answer is correct
The correct answer is A. (2) पर खाली वृत्त और दाईं ओर छायांकन / Open circle at (2) and shading right. Solving gives (x>2). A strict inequality excludes the boundary so an open circle is used.
Step 3
Exam Tip
हल करने पर (x>2) मिलता है। कठोर असमता में सीमा शामिल नहीं होती इसलिए खाली वृत्त बनता है।
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असमानता \(-5x+3\le 18\) का संख्या रेखा पर सही अंतराल कौन सा है?
Which interval is correct on the number line for \(-5x+3\le 18\)?
#linear-inequalities
#negative-division
#interval-form
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A (\(-\infty,-3]\)
B (\(-3,\infty\))
C \([-3,\infty\))
D (\(-\infty,-3\))
Explanation opens after your attempt
Correct Answer
C. \([-3,\infty\))
Step 1
Concept
Solving gives \(-5x\le 15\) and then \(x\ge -3\). The inequality sign reverses when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is C. \([-3,\infty\)). Solving gives \(-5x\le 15\) and then \(x\ge -3\). The inequality sign reverses when dividing by a negative.
Step 3
Exam Tip
हल करने पर \(-5x\le 15\) और फिर \(x\ge -3\) मिलता है। ऋणात्मक से भाग देने पर असमता का चिह्न बदलता है।
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संख्या रेखा पर \(x\in[-6,4\)\cap(-2,8]) का समान भाग कौन सा है?
What is the common part of \(x\in[-6,4\)\cap(-2,8]) on the number line?
#linear-inequalities
#intersection
#interval-operation
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A ([-6,8])
B ((-2,4))
C ([-2,4])
D ((-6,8))
Explanation opens after your attempt
Correct Answer
B. ((-2,4))
Step 1
Concept
The common part is greater than (-2) and less than (4). Both endpoints are excluded in this intersection.
Step 2
Why this answer is correct
The correct answer is B. ((-2,4)). The common part is greater than (-2) and less than (4). Both endpoints are excluded in this intersection.
Step 3
Exam Tip
दोनों अंतरालों का समान भाग (-2) से बड़ा और (4) से छोटा है। दोनों सीमाएँ इस समान भाग में शामिल नहीं हैं।
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यदि (x\in\(-\infty,-1\)\cup[-1,5)), तो संख्या रेखा पर सरल अंतराल क्या होगा?
If (x\in\(-\infty,-1\)\cup[-1,5)), what will be the simplified interval on the number line?
#linear-inequalities
#union-simplification
#number-line
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A (\(-\infty,5\))
B (\(-\infty,5]\)
C ((-1,5))
D ([ -1,5 ))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,5\))
Step 1
Concept
The second interval includes (-1), so no gap remains. The union gives all numbers before (5).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,5\)). The second interval includes (-1), so no gap remains. The union gives all numbers before (5).
Step 3
Exam Tip
दूसरा अंतराल (-1) को शामिल करता है इसलिए बीच में कोई अंतर नहीं बचता। संयुक्त भाग (5) से पहले तक की सभी संख्याएँ देता है।
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यदि (x) वास्तविक संख्या है और \(x\notin[-4,2\)), तो संख्या रेखा पर सही अंतराल कौन सा है?
If (x) is a real number and \(x\notin[-4,2\)), which interval is correct on the number line?
#linear-inequalities
#complement
#number-line
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A ((-\infty,-4]\cup\(2,\infty\))
B (\(-\infty,-4\)\cup[2,\infty))
C ([-4,2))
D (\(-\infty,-4\)\cup\(2,\infty\))
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,-4\)\cup[2,\infty))
Step 1
Concept
The original interval includes (-4) but excludes (2). In the complement, (-4) is excluded and (2) is included.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,-4\)\cup[2,\infty)). The original interval includes (-4) but excludes (2). In the complement, (-4) is excluded and (2) is included.
Step 3
Exam Tip
मूल अंतराल में (-4) शामिल है लेकिन (2) शामिल नहीं है। पूरक में (-4) बाहर और (2) शामिल होगा।
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संयुक्त असमानता \(1\le \frac{2x-3}{5}<3\) का संख्या रेखा अंतराल क्या है?
What is the number line interval for the compound inequality \(1\le \frac{2x-3}{5}<3\)?
#linear-inequalities
#compound
#fraction
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A ((4,9])
B ([4,9))
C ((4,9))
D ([4,9])
Explanation opens after your attempt
Correct Answer
B. ([4,9))
Step 1
Concept
Solving gives \(5\le 2x-3<15\) and then \(4\le x<9\). Therefore (4) is included but (9) is excluded.
Step 2
Why this answer is correct
The correct answer is B. ([4,9)). Solving gives \(5\le 2x-3<15\) and then \(4\le x<9\). Therefore (4) is included but (9) is excluded.
Step 3
Exam Tip
हल करने पर \(5\le 2x-3<15\) और फिर \(4\le x<9\) मिलता है। इसलिए (4) शामिल है लेकिन (9) शामिल नहीं है।
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यदि (x) पूर्णांक है और \(-4\le x<3\), तो संख्या रेखा पर कितने पूर्णांक बिंदु चिह्नित होंगे?
If (x) is an integer and \(-4\le x<3\), how many integer points will be marked on the number line?
#linear-inequalities
#integer-count
#number-line
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A (6)
B (7)
C (8)
D (5)
Explanation opens after your attempt
Step 1
Concept
The integers are (-4,-3,-2,-1,0,1,2). The endpoint (3) is open, so it is not counted.
Step 2
Why this answer is correct
The correct answer is B. (7). The integers are (-4,-3,-2,-1,0,1,2). The endpoint (3) is open, so it is not counted.
Step 3
Exam Tip
पूर्णांक (-4,-3,-2,-1,0,1,2) हैं। (3) खुला सिरा है इसलिए उसे नहीं गिनते।
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