यदि ( -3x+7>16 ) है तो (x) के लिए सही अंतराल कौन सा है?
If ( -3x+7>16 ), which interval is correct for (x)?
#linear-inequalities
#introduction
#sign-reversal
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A (x<-3)
B (x>-3)
C (x<3)
D (x>3)
Explanation opens after your attempt
Step 1
Concept
Dividing by a negative number reverses the inequality sign. In exams, always reverse the sign when dividing by (-3).
Step 2
Why this answer is correct
The correct answer is A. (x<-3). Dividing by a negative number reverses the inequality sign. In exams, always reverse the sign when dividing by (-3).
Step 3
Exam Tip
ऋणात्मक संख्या से भाग देने पर असमता का चिन्ह बदलता है। परीक्षा में (-3) से भाग देते समय चिन्ह अवश्य पलटें।
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यदि \(x\in\mathbb{R}\) और \(\frac{2x-1}{3}\leq \frac{x+5}{6}\) है तो (x) का सही हल कौन सा है?
If \(x\in\mathbb{R}\) and \(\frac{2x-1}{3}\leq \frac{x+5}{6}\), which solution for (x) is correct?
#fractional-inequality
#lcm
#linear-inequality
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A \(x\leq \frac{7}{3}\)
B \(x\geq \frac{7}{3}\)
C \(x\leq -\frac{7}{3}\)
D \(x\geq -\frac{7}{3}\)
Explanation opens after your attempt
Correct Answer
A. \(x\leq \frac{7}{3}\)
Step 1
Concept
Multiplying by (6) gives \(4x-2\leq x+5\), so \(3x\leq 7\). Multiplying by a positive LCM does not change the inequality sign.
Step 2
Why this answer is correct
The correct answer is A. \(x\leq \frac{7}{3}\). Multiplying by (6) gives \(4x-2\leq x+5\), so \(3x\leq 7\). Multiplying by a positive LCM does not change the inequality sign.
Step 3
Exam Tip
(6) से गुणा करने पर \(4x-2\leq x+5\) और \(3x\leq 7\) मिलता है। धनात्मक लघुत्तम समापवर्त्य से गुणा करते समय असमता का चिन्ह नहीं बदलता।
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किस असमता का हल समुच्चय वास्तविक संख्याओं में (\(-4,\infty\)) है?
Which inequality has solution set (\(-4,\infty\)) in real numbers?
#solution-set
#interval
#linear-inequality
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A (2x+8>0)
B (2x+8<0)
C (2x-8>0)
D (2x-8<0)
Explanation opens after your attempt
Correct Answer
A. (2x+8>0)
Step 1
Concept
From (2x+8>0), we get (x>-4). Read the left endpoint of the interval carefully.
Step 2
Why this answer is correct
The correct answer is A. (2x+8>0). From (2x+8>0), we get (x>-4). Read the left endpoint of the interval carefully.
Step 3
Exam Tip
(2x+8>0) से (x>-4) मिलता है। अंतराल की बाईं सीमा को ध्यान से पढ़ें।
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असमता (-2(4x-3)\geq 10) का हल कौन सा है?
What is the solution of (-2(4x-3)\geq 10)?
#bracket-inequality
#negative-division
#linear
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A \(x\leq -\frac{1}{2}\)
B \(x\geq -\frac{1}{2}\)
C \(x\leq \frac{1}{2}\)
D \(x\geq \frac{1}{2}\)
Explanation opens after your attempt
Correct Answer
A. \(x\leq -\frac{1}{2}\)
Step 1
Concept
Simplifying gives \(-8x+6\geq 10\), then \(-8x\geq 4\), so \(x\leq -\frac{1}{2}\). Reverse the sign when dividing by a negative coefficient.
Step 2
Why this answer is correct
The correct answer is A. \(x\leq -\frac{1}{2}\). Simplifying gives \(-8x+6\geq 10\), then \(-8x\geq 4\), so \(x\leq -\frac{1}{2}\). Reverse the sign when dividing by a negative coefficient.
Step 3
Exam Tip
सरलीकरण से \(-8x+6\geq 10\), फिर \(-8x\geq 4\) और \(x\leq -\frac{1}{2}\) मिलता है। ऋणात्मक गुणांक से भाग देते समय चिन्ह पलटें।
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यदि (x) एक पूर्णांक है और \(2x-5\leq 9\) है तो सबसे बड़ा संभव (x) क्या है?
If (x) is an integer and \(2x-5\leq 9\), what is the greatest possible (x)?
#integer-solution
#greatest-value
#linear-inequality
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A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The inequality gives \(2x\leq 14\), so \(x\leq 7\). With an integer restriction, choose an integer final answer.
Step 2
Why this answer is correct
The correct answer is B. (7). The inequality gives \(2x\leq 14\), so \(x\leq 7\). With an integer restriction, choose an integer final answer.
Step 3
Exam Tip
असमता से \(2x\leq 14\) इसलिए \(x\leq 7\) मिलता है। पूर्णांक प्रतिबंध हो तो अंतिम उत्तर पूर्णांक ही लें।
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यदि \(x\in\mathbb{Z}\) और \(\frac{x+5}{2}>3\) है तो सबसे छोटा संभव (x) क्या है?
If \(x\in\mathbb{Z}\) and \(\frac{x+5}{2}>3\), what is the least possible (x)?
#integer-solution
#fractional-inequality
#least-value
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A (2)
B (1)
C (3)
D (6)
Explanation opens after your attempt
Step 1
Concept
Multiplying by (2) gives (x+5>6), so (x>1). For integer solutions, the least value is (2).
Step 2
Why this answer is correct
The correct answer is A. (2). Multiplying by (2) gives (x+5>6), so (x>1). For integer solutions, the least value is (2).
Step 3
Exam Tip
(2) से गुणा करने पर (x+5>6) और (x>1) मिलता है। पूर्णांक हल में सबसे छोटा मान (2) होगा।
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यदि \(x\in\mathbb{R}\) और (5-2x<1) है तो सही हल कौन सा है?
If \(x\in\mathbb{R}\) and (5-2x<1), which solution is correct?
#real-solution
#sign-change
#linear-inequality
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A (x>2)
B (x<2)
C (x>-2)
D (x< -2)
Explanation opens after your attempt
Step 1
Concept
From (5-2x<1), we get (-2x<-4), hence (x>2). Reversing the sign for a negative coefficient is the key step.
Step 2
Why this answer is correct
The correct answer is A. (x>2). From (5-2x<1), we get (-2x<-4), hence (x>2). Reversing the sign for a negative coefficient is the key step.
Step 3
Exam Tip
(5-2x<1) से (-2x<-4) और (x>2) मिलता है। ऋणात्मक गुणांक पर चिन्ह पलटना मुख्य चरण है।
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निम्न में से कौन सी क्रिया असमता (a>b) को हमेशा (ac>bc) में बदलने के लिए वैध है?
Which operation always changes inequality (a>b) into (ac>bc)?
#properties-of-inequality
#multiplication
#concept
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A (c>0)
B (c<0)
C (c=0)
D \(c\neq 0\)
Explanation opens after your attempt
Step 1
Concept
Multiplying by a positive number keeps the direction of inequality unchanged. If (c<0), the sign reverses.
Step 2
Why this answer is correct
The correct answer is A. (c>0). Multiplying by a positive number keeps the direction of inequality unchanged. If (c<0), the sign reverses.
Step 3
Exam Tip
धनात्मक संख्या से गुणा करने पर असमता की दिशा वही रहती है। (c<0) होने पर चिन्ह पलट जाता है।
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असमता (3(x-2)\geq 2x+5) का हल कौन सा है?
What is the solution of (3(x-2)\geq 2x+5)?
#linear-inequality
#brackets
#solution
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A \(x\geq 11\)
B \(x\leq 11\)
C \(x\geq -11\)
D \(x\leq -11\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq 11\)
Step 1
Concept
Simplifying gives \(3x-6\geq 2x+5\), so \(x\geq 11\). While opening brackets, handle signs and constants carefully.
Step 2
Why this answer is correct
The correct answer is A. \(x\geq 11\). Simplifying gives \(3x-6\geq 2x+5\), so \(x\geq 11\). While opening brackets, handle signs and constants carefully.
Step 3
Exam Tip
सरलीकरण से \(3x-6\geq 2x+5\) इसलिए \(x\geq 11\) है। कोष्ठक खोलते समय चिन्ह और स्थिर पद सावधानी से लिखें।
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यदि (4x+1<2x+9) है तो (x) का सही अंतराल कौन सा है?
If (4x+1<2x+9), which interval for (x) is correct?
#interval-notation
#linear-inequality
#real-numbers
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A (\(-\infty,4\))
B (\(4,\infty\))
C (\(-\infty,-4\))
D (\(-4,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,4\))
Step 1
Concept
From (4x+1<2x+9), (2x<8), so (x<4). Move variable terms to one side correctly.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,4\)). From (4x+1<2x+9), (2x<8), so (x<4). Move variable terms to one side correctly.
Step 3
Exam Tip
(4x+1<2x+9) से (2x<8) इसलिए (x<4) है। चर वाले पदों को एक ओर सही ढंग से लाएं।
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असमता \(\frac{x-1}{3}\leq 2\) का हल कौन सा है?
What is the solution of \(\frac{x-1}{3}\leq 2\)?
#fraction-inequality
#positive-multiplier
#solution
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A \(x\leq 7\)
B \(x\geq 7\)
C \(x\leq 5\)
D \(x\geq 5\)
Explanation opens after your attempt
Correct Answer
A. \(x\leq 7\)
Step 1
Concept
Multiplying by positive (3) does not change the sign and gives \(x-1\leq 6\). If the denominator is positive, the direction stays the same.
Step 2
Why this answer is correct
The correct answer is A. \(x\leq 7\). Multiplying by positive (3) does not change the sign and gives \(x-1\leq 6\). If the denominator is positive, the direction stays the same.
Step 3
Exam Tip
धनात्मक (3) से गुणा करने पर चिन्ह नहीं बदलता और \(x-1\leq 6\) मिलता है। हर धनात्मक हो तो दिशा वही रहती है।
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असमता \(\frac{2-x}{4}>1\) का हल कौन सा है?
What is the solution of \(\frac{2-x}{4}>1\)?
#fraction-inequality
#negative-coefficient
#hard
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A (x<-2)
B (x> -2)
C (x<2)
D (x>2)
Explanation opens after your attempt
Step 1
Concept
From (2-x>4), we get (-x>2), hence (x<-2). When converting (-x) to (x), the sign reverses.
Step 2
Why this answer is correct
The correct answer is A. (x<-2). From (2-x>4), we get (-x>2), hence (x<-2). When converting (-x) to (x), the sign reverses.
Step 3
Exam Tip
(2-x>4) से (-x>2) और (x<-2) मिलता है। (-x) को (x) बनाने पर चिन्ह पलटता है।
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किस असमता का हल \(x\leq -1\) है?
Which inequality has solution \(x\leq -1\)?
#reverse-sign
#identify-inequality
#linear
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A \(-2x\geq 2\)
B \(-2x\leq 2\)
C \(2x\geq -2\)
D (2x<-2)
Explanation opens after your attempt
Correct Answer
A. \(-2x\geq 2\)
Step 1
Concept
Dividing \(-2x\geq 2\) by (-2) gives \(x\leq -1\). Reverse the sign during negative division.
Step 2
Why this answer is correct
The correct answer is A. \(-2x\geq 2\). Dividing \(-2x\geq 2\) by (-2) gives \(x\leq -1\). Reverse the sign during negative division.
Step 3
Exam Tip
\(-2x\geq 2\) को (-2) से भाग देने पर \(x\leq -1\) मिलता है। ऋणात्मक विभाजन में चिन्ह उल्टा करें।
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यदि \(x\in\mathbb{N}\) और (3x+2<20) है तो हल समुच्चय कौन सा है?
If \(x\in\mathbb{N}\) and (3x+2<20), what is the solution set?
#natural-numbers
#solution-set
#introduction
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A ({1,2,3,4,5})
B ({0,1,2,3,4,5})
C ({1,2,3,4,5,6})
D ({2,3,4,5,6})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,4,5})
Step 1
Concept
The inequality gives (3x<18), so (x<6). In \(\mathbb{N}\), usually take positive integers starting from (1).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4,5}). The inequality gives (3x<18), so (x<6). In \(\mathbb{N}\), usually take positive integers starting from (1).
Step 3
Exam Tip
असमता से (3x<18) इसलिए (x<6) है। \(\mathbb{N}\) में सामान्यतः (1) से शुरू होने वाले धनात्मक पूर्णांक लें।
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यदि \(x\in\mathbb{Z}\) और \(-1<2x+3\leq 9\) है तो हल समुच्चय कौन सा है?
If \(x\in\mathbb{Z}\) and \(-1<2x+3\leq 9\), what is the solution set?
#compound-inequality
#integers
#solution-set
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A ({-1,0,1,2,3})
B ({-2,-1,0,1,2,3})
C ({0,1,2,3})
D ({-1,0,1,2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({-1,0,1,2,3})
Step 1
Concept
From \(-1<2x+3\leq 9\), we get \(-4<2x\leq 6\), hence \(-2<x\leq 3\). For integers, handle open and closed endpoints carefully.
Step 2
Why this answer is correct
The correct answer is A. ({-1,0,1,2,3}). From \(-1<2x+3\leq 9\), we get \(-4<2x\leq 6\), hence \(-2<x\leq 3\). For integers, handle open and closed endpoints carefully.
Step 3
Exam Tip
\(-1<2x+3\leq 9\) से \(-4<2x\leq 6\) और \(-2<x\leq 3\) मिलता है। पूर्णांक चुनते समय खुले और बंद सिरों का ध्यान रखें।
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अंतराल ([-2,5)) को असमता के रूप में कैसे लिखेंगे?
How is interval ([-2,5)) written as an inequality?
#interval-to-inequality
#notation
#concept
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A \(-2\leq x<5\)
B \(-2<x\leq 5\)
C (-2<x<5)
D \(-2\leq x\leq 5\)
Explanation opens after your attempt
Correct Answer
A. \(-2\leq x<5\)
Step 1
Concept
In ([-2,5)), (-2) is included and (5) is not included. A square bracket includes equality.
Step 2
Why this answer is correct
The correct answer is A. \(-2\leq x<5\). In ([-2,5)), (-2) is included and (5) is not included. A square bracket includes equality.
Step 3
Exam Tip
([-2,5)) में (-2) शामिल है और (5) शामिल नहीं है। वर्ग कोष्ठक के लिए बराबरी शामिल होती है।
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असमता \(7-3x\leq -2\) का हल कौन सा है?
What is the solution of \(7-3x\leq -2\)?
#linear-inequality
#negative-division
#solution
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A \(x\geq 3\)
B \(x\leq 3\)
C \(x\geq -3\)
D \(x\leq -3\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq 3\)
Step 1
Concept
From \(7-3x\leq -2\), \(-3x\leq -9\), so \(x\geq 3\). Dividing by a negative coefficient reverses the sign.
Step 2
Why this answer is correct
The correct answer is A. \(x\geq 3\). From \(7-3x\leq -2\), \(-3x\leq -9\), so \(x\geq 3\). Dividing by a negative coefficient reverses the sign.
Step 3
Exam Tip
\(7-3x\leq -2\) से \(-3x\leq -9\) और \(x\geq 3\) मिलता है। ऋणात्मक गुणांक से भाग देने पर चिन्ह पलटता है।
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कौन सा मान (2x-7>5) का हल नहीं है?
Which value is not a solution of (2x-7>5)?
#boundary-value
#not-solution
#strict-inequality
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A (x=6)
B (x=7)
C (x=8)
D (x=9)
Explanation opens after your attempt
Step 1
Concept
The inequality gives (2x>12), so (x>6). The boundary value (x=6) is not included because the sign is strict.
Step 2
Why this answer is correct
The correct answer is A. (x=6). The inequality gives (2x>12), so (x>6). The boundary value (x=6) is not included because the sign is strict.
Step 3
Exam Tip
असमता से (2x>12) इसलिए (x>6) है। सीमा (x=6) खुले चिन्ह के कारण हल नहीं है।
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किस स्थिति में (a+c<b+c) निष्कर्ष (a<b) से हमेशा निकलेगा?
In which situation does (a+c<b+c) always follow from (a<b)?
#addition-property
#real-number
#concept
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A \(c\in\mathbb{R}\)
B (c>0)
C (c<0)
D \(c\neq 0\)
Explanation opens after your attempt
Correct Answer
A. \(c\in\mathbb{R}\)
Step 1
Concept
Adding the same real number does not change the direction of inequality. For addition, the sign of (c) does not matter.
Step 2
Why this answer is correct
The correct answer is A. \(c\in\mathbb{R}\). Adding the same real number does not change the direction of inequality. For addition, the sign of (c) does not matter.
Step 3
Exam Tip
एक ही वास्तविक संख्या जोड़ने से असमता की दिशा नहीं बदलती। जोड़ के लिए (c) का धनात्मक या ऋणात्मक होना मायने नहीं रखता।
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असमता (2(3x-1)<5x+6) का हल कौन सा है?
What is the solution of (2(3x-1)<5x+6)?
#bracket-expansion
#linear-inequality
#solution
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A (x<8)
B (x>8)
C (x<4)
D (x>4)
Explanation opens after your attempt
Step 1
Concept
From (6x-2<5x+6), we get (x<8). When variables are on both sides, separate like terms first.
Step 2
Why this answer is correct
The correct answer is A. (x<8). From (6x-2<5x+6), we get (x<8). When variables are on both sides, separate like terms first.
Step 3
Exam Tip
(6x-2<5x+6) से (x<8) मिलता है। दोनों ओर चर हों तो पहले समान पदों को अलग करें।
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यदि \(x\in\mathbb{R}\) और \(0\leq 4x-8<12\) है तो (x) का अंतराल क्या है?
If \(x\in\mathbb{R}\) and \(0\leq 4x-8<12\), what is the interval for (x)?
#compound-inequality
#interval
#real-solution
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A ([2,5))
B ((2,5])
C ([2,5])
D ((2,5))
Explanation opens after your attempt
Correct Answer
A. ([2,5))
Step 1
Concept
From \(0\leq 4x-8<12\), we get \(8\leq 4x<20\), hence \(2\leq x<5\). Solve both ends of a compound inequality together.
Step 2
Why this answer is correct
The correct answer is A. ([2,5)). From \(0\leq 4x-8<12\), we get \(8\leq 4x<20\), hence \(2\leq x<5\). Solve both ends of a compound inequality together.
Step 3
Exam Tip
\(0\leq 4x-8<12\) से \(8\leq 4x<20\) और \(2\leq x<5\) मिलता है। संयुक्त असमता में दोनों सिरों को साथ-साथ हल करें।
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यदि \(\frac{x+2}{5}\geq \frac{x-1}{2}\) है तो सही हल कौन सा है?
If \(\frac{x+2}{5}\geq \frac{x-1}{2}\), which solution is correct?
#fractional-inequality
#lcm
#linear
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A \(x\leq 3\)
B \(x\geq 3\)
C \(x\leq -3\)
D \(x\geq -3\)
Explanation opens after your attempt
Correct Answer
A. \(x\leq 3\)
Step 1
Concept
Multiplying by (10) gives \(2x+4\geq 5x-5\), so \(9\geq 3x\). Multiplication by a positive LCM does not change the sign.
Step 2
Why this answer is correct
The correct answer is A. \(x\leq 3\). Multiplying by (10) gives \(2x+4\geq 5x-5\), so \(9\geq 3x\). Multiplication by a positive LCM does not change the sign.
Step 3
Exam Tip
(10) से गुणा करने पर \(2x+4\geq 5x-5\) और \(9\geq 3x\) मिलता है। धनात्मक लघुत्तम समापवर्त्य से गुणा करने पर चिन्ह नहीं बदलता।
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कौन सा विकल्प \(x\geq -2\) और (x<3) का संयुक्त हल है?
Which option is the combined solution of \(x\geq -2\) and (x<3)?
#intersection
#compound-solution
#interval
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A ([-2,3))
B ((-2,3])
C ((-2,3))
D ([-2,3])
Explanation opens after your attempt
Correct Answer
A. ([-2,3))
Step 1
Concept
Taking both conditions together gives \(-2\leq x<3\). The combined solution is the intersection of both inequalities.
Step 2
Why this answer is correct
The correct answer is A. ([-2,3)). Taking both conditions together gives \(-2\leq x<3\). The combined solution is the intersection of both inequalities.
Step 3
Exam Tip
दोनों शर्तों को साथ लेने पर \(-2\leq x<3\) मिलता है। संयुक्त हल में दोनों असमताओं का प्रतिच्छेद लिया जाता है।
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कौन सा विकल्प (x< -1) या \(x\geq 4\) का सही अंतराल रूप है?
Which option is the correct interval form of (x<-1) or \(x\geq 4\)?
#union
#interval-notation
#compound-inequality
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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
A. (\(-\infty,-1\)\cup[4,\infty))
Step 1
Concept
Or means the union of the two solution sets. Match open and closed endpoints with the inequality signs.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,-1\)\cup[4,\infty)). Or means the union of the two solution sets. Match open and closed endpoints with the inequality signs.
Step 3
Exam Tip
या का अर्थ दोनों हलों का संघ है। खुले और बंद सिरों को असमता के चिन्ह से मिलाएं।
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यदि (2x+3) का मान (11) से कम और (1) से अधिक है तो (x) का हल कौन सा है?
If (2x+3) is less than (11) and greater than (1), what is the solution for (x)?
#word-problem
#compound-inequality
#translation
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A (-1<x<4)
B \(-1\leq x<4\)
C \(-1<x\leq 4\)
D (-4<x<1)
Explanation opens after your attempt
Correct Answer
A. (-1<x<4)
Step 1
Concept
From (1<2x+3<11), we get (-2<2x<8), hence (-1<x<4). First convert the statement into a compound inequality.
Step 2
Why this answer is correct
The correct answer is A. (-1<x<4). From (1<2x+3<11), we get (-2<2x<8), hence (-1<x<4). First convert the statement into a compound inequality.
Step 3
Exam Tip
(1<2x+3<11) से (-2<2x<8) और (-1<x<4) मिलता है। वाक्य को पहले संयुक्त असमता में बदलें।
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यदि \(x\in\mathbb{Z}\) और \(-5\leq x<2\) है तो कितने पूर्णांक हल हैं?
If \(x\in\mathbb{Z}\) and \(-5\leq x<2\), how many integer solutions are there?
#counting-integers
#interval
#solution-set
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A (7)
B (6)
C (8)
D (5)
Explanation opens after your attempt
Step 1
Concept
The solutions are (-5,-4,-3,-2,-1,0,1), so there are (7). The open endpoint (2) is not included.
Step 2
Why this answer is correct
The correct answer is A. (7). The solutions are (-5,-4,-3,-2,-1,0,1), so there are (7). The open endpoint (2) is not included.
Step 3
Exam Tip
हल (-5,-4,-3,-2,-1,0,1) हैं इसलिए कुल (7) हैं। खुले सिरे पर (2) शामिल नहीं है।
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असमता \(9\leq 3x+6\) का सबसे सरल रूप कौन सा है?
What is the simplest form of \(9\leq 3x+6\)?
#reverse-reading
#linear-inequality
#simplification
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A \(x\geq 1\)
B \(x\leq 1\)
C \(x\geq 5\)
D \(x\leq 5\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq 1\)
Step 1
Concept
From \(9\leq 3x+6\), we get \(3\leq 3x\), hence \(x\geq 1\). Read the meaning carefully when the inequality is written in reverse order.
Step 2
Why this answer is correct
The correct answer is A. \(x\geq 1\). From \(9\leq 3x+6\), we get \(3\leq 3x\), hence \(x\geq 1\). Read the meaning carefully when the inequality is written in reverse order.
Step 3
Exam Tip
\(9\leq 3x+6\) से \(3\leq 3x\) और \(x\geq 1\) मिलता है। असमता उलटी लिखी हो तो भी अर्थ ध्यान से पढ़ें।
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यदि (x) वास्तविक है तो (-4x-1<7) का हल कौन सा है?
If (x) is real, what is the solution of (-4x-1<7)?
#negative-coefficient
#real-solution
#linear
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A (x>-2)
B (x<-2)
C (x>2)
D (x<2)
Explanation opens after your attempt
Step 1
Concept
From (-4x<8), we get (x>-2). Dividing by negative (-4) reverses the sign.
Step 2
Why this answer is correct
The correct answer is A. (x>-2). From (-4x<8), we get (x>-2). Dividing by negative (-4) reverses the sign.
Step 3
Exam Tip
(-4x<8) से (x>-2) मिलता है। ऋणात्मक (-4) से भाग देने पर चिन्ह पलटता है।
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कौन सा कथन \(x\leq 5\) के बराबर है?
Which statement is equivalent to \(x\leq 5\)?
#equivalent-inequality
#notation
#concept
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A \(5\geq x\)
B (5<x)
C (x>5)
D \(5\leq x\)
Explanation opens after your attempt
Correct Answer
A. \(5\geq x\)
Step 1
Concept
Writing \(x\leq 5\) in reversed order gives \(5\geq x\). When switching sides, the visible direction of the sign also switches.
Step 2
Why this answer is correct
The correct answer is A. \(5\geq x\). Writing \(x\leq 5\) in reversed order gives \(5\geq x\). When switching sides, the visible direction of the sign also switches.
Step 3
Exam Tip
\(x\leq 5\) को उलटकर लिखने पर \(5\geq x\) होता है। पक्ष बदलते समय चिन्ह की दिशा भी उलटी दिखती है।
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यदि \(x\in\mathbb{R}\) और (x-3<0) है तो सही संख्या रेखा वर्णन कौन सा है?
If \(x\in\mathbb{R}\) and (x-3<0), which number-line description is correct?
#number-line
#strict-inequality
#representation
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A (3) पर खुला वृत्त और बाईं ओर छाया
B (3) पर बंद वृत्त और बाईं ओर छाया
C (3) पर खुला वृत्त और दाईं ओर छाया
D (3) पर बंद वृत्त और दाईं ओर छाया
Explanation opens after your attempt
Correct Answer
A. (3) पर खुला वृत्त और बाईं ओर छाया
Step 1
Concept
In (x<3), (3) is not included and smaller numbers are included. Use an open circle for a strict inequality.
Step 2
Why this answer is correct
The correct answer is A. (3) पर खुला वृत्त और बाईं ओर छाया. In (x<3), (3) is not included and smaller numbers are included. Use an open circle for a strict inequality.
Step 3
Exam Tip
(x<3) में (3) शामिल नहीं है और उससे छोटी संख्याएं शामिल हैं। सख्त असमता के लिए खुला वृत्त बनाएं।
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असमता \(3x+4\geq x-6\) का हल कौन सा है?
What is the solution of \(3x+4\geq x-6\)?
#linear-inequality
#variable-both-sides
#solution
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A \(x\geq -5\)
B \(x\leq -5\)
C \(x\geq 5\)
D \(x\leq 5\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq -5\)
Step 1
Concept
From \(3x+4\geq x-6\), \(2x\geq -10\), so \(x\geq -5\). Keeping like terms organized reduces mistakes.
Step 2
Why this answer is correct
The correct answer is A. \(x\geq -5\). From \(3x+4\geq x-6\), \(2x\geq -10\), so \(x\geq -5\). Keeping like terms organized reduces mistakes.
Step 3
Exam Tip
\(3x+4\geq x-6\) से \(2x\geq -10\) और \(x\geq -5\) मिलता है। समान पदों को व्यवस्थित रखकर गलती कम होती है।
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यदि (p>q) है तो निम्न में से कौन सा निष्कर्ष हमेशा सही है?
If (p>q), which conclusion is always correct?
#inequality-properties
#always-true
#reasoning
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A (p-7>q-7)
B (7-p>7-q)
C \(p^2>q^2\)
D \(\frac{1}{p}>\frac{1}{q}\)
Explanation opens after your attempt
Correct Answer
A. (p-7>q-7)
Step 1
Concept
Subtracting the same number from both sides does not change the direction. Squares and reciprocals need extra conditions.
Step 2
Why this answer is correct
The correct answer is A. (p-7>q-7). Subtracting the same number from both sides does not change the direction. Squares and reciprocals need extra conditions.
Step 3
Exam Tip
दोनों ओर समान संख्या घटाने से असमता की दिशा नहीं बदलती। वर्ग और व्युत्क्रम के लिए अतिरिक्त शर्तें चाहिए।
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यदि \(x\in\mathbb{Z}\) और (4x-1>15) है तो सबसे छोटा संभव (x) क्या है?
If \(x\in\mathbb{Z}\) and (4x-1>15), what is the least possible (x)?
#least-integer
#strict-inequality
#solution
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A (5)
B (4)
C (6)
D (3)
Explanation opens after your attempt
Step 1
Concept
From (4x>16), (x>4), so the least integer is (5). A strict sign does not include the boundary value.
Step 2
Why this answer is correct
The correct answer is A. (5). From (4x>16), (x>4), so the least integer is (5). A strict sign does not include the boundary value.
Step 3
Exam Tip
(4x>16) से (x>4) है इसलिए सबसे छोटा पूर्णांक (5) है। सख्त चिन्ह में सीमा मान शामिल नहीं होता।
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किस असमता का हल (\(-\infty,2]\) है?
Which inequality has solution (\(-\infty,2]\)?
#identify-inequality
#interval
#closed-endpoint
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A \(3x-6\leq 0\)
B (3x-6<0)
C \(3x-6\geq 0\)
D (3x-6>0)
Explanation opens after your attempt
Correct Answer
A. \(3x-6\leq 0\)
Step 1
Concept
From \(3x-6\leq 0\), we get \(x\leq 2\). The closed right endpoint comes from the equality sign.
Step 2
Why this answer is correct
The correct answer is A. \(3x-6\leq 0\). From \(3x-6\leq 0\), we get \(x\leq 2\). The closed right endpoint comes from the equality sign.
Step 3
Exam Tip
\(3x-6\leq 0\) से \(x\leq 2\) मिलता है। बंद दायां सिरा बराबरी वाले चिन्ह से आता है।
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यदि (x) वास्तविक है और \(2x+1\geq 7\) तथा (x-4<3) है तो संयुक्त हल क्या है?
If (x) is real and \(2x+1\geq 7\) and (x-4<3), what is the combined solution?
#system-of-inequalities
#intersection
#real-solution
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A \(3\leq x<7\)
B \(3<x\leq 7\)
C \(x\geq 7\)
D (x<3)
Explanation opens after your attempt
Correct Answer
A. \(3\leq x<7\)
Step 1
Concept
The first inequality gives \(x\geq 3\), and the second gives (x<7). Their intersection is \(3\leq x<7\).
Step 2
Why this answer is correct
The correct answer is A. \(3\leq x<7\). The first inequality gives \(x\geq 3\), and the second gives (x<7). Their intersection is \(3\leq x<7\).
Step 3
Exam Tip
पहली असमता से \(x\geq 3\) और दूसरी से (x<7) मिलता है। दोनों शर्तों का प्रतिच्छेद \(3\leq x<7\) है।
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असमता (8-2(3x+1)\leq 0) का हल कौन सा है?
What is the solution of (8-2(3x+1)\leq 0)?
#brackets
#negative-multiplier
#linear-inequality
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A \(x\geq 1\)
B \(x\leq 1\)
C \(x\geq -1\)
D \(x\leq -1\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq 1\)
Step 1
Concept
Simplifying gives \(8-6x-2\leq 0\), that is \(6-6x\leq 0\). Then \(-6x\leq -6\) gives \(x\geq 1\).
Step 2
Why this answer is correct
The correct answer is A. \(x\geq 1\). Simplifying gives \(8-6x-2\leq 0\), that is \(6-6x\leq 0\). Then \(-6x\leq -6\) gives \(x\geq 1\).
Step 3
Exam Tip
सरलीकरण से \(8-6x-2\leq 0\) यानी \(6-6x\leq 0\) मिलता है। फिर \(-6x\leq -6\) से \(x\geq 1\) है।
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कौन सा विकल्प (x=2) को हल में शामिल करता है लेकिन (x=6) को शामिल नहीं करता?
Which option includes (x=2) in the solution but excludes (x=6)?
#endpoint-inclusion
#compound-inequality
#concept
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A \(2\leq x<6\)
B \(2<x\leq 6\)
C (2<x<6)
D \(2\leq x\leq 6\)
Explanation opens after your attempt
Correct Answer
A. \(2\leq x<6\)
Step 1
Concept
In \(2\leq x<6\), (2) is included and (6) is excluded. Check the equality sign for inclusion.
Step 2
Why this answer is correct
The correct answer is A. \(2\leq x<6\). In \(2\leq x<6\), (2) is included and (6) is excluded. Check the equality sign for inclusion.
Step 3
Exam Tip
\(2\leq x<6\) में (2) शामिल है और (6) शामिल नहीं है। समावेशन के लिए बराबरी का चिन्ह देखें।
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यदि (-2a<6) है तो (a) के लिए सही निष्कर्ष क्या है?
If (-2a<6), what is the correct conclusion for (a)?
#negative-division
#basic-property
#linear
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A (a>-3)
B (a<-3)
C (a>3)
D (a<3)
Explanation opens after your attempt
Step 1
Concept
Dividing by (-2) reverses the inequality to (a>-3). Division by a negative number is the most common mistake point.
Step 2
Why this answer is correct
The correct answer is A. (a>-3). Dividing by (-2) reverses the inequality to (a>-3). Division by a negative number is the most common mistake point.
Step 3
Exam Tip
(-2) से भाग देने पर असमता उलटकर (a>-3) हो जाती है। ऋणात्मक संख्या से भाग देना सबसे सामान्य गलती वाला चरण है।
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निम्न में से कौन सा कथन \(x\in[1,4]\) का सही अर्थ है?
Which statement correctly means \(x\in[1,4]\)?
#closed-interval
#notation
#inequality-form
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A \(1\leq x\leq 4\)
B (1<x<4)
C \(1\leq x<4\)
D \(1<x\leq 4\)
Explanation opens after your attempt
Correct Answer
A. \(1\leq x\leq 4\)
Step 1
Concept
The closed interval ([1,4]) includes both endpoints. Therefore equality appears on both sides.
Step 2
Why this answer is correct
The correct answer is A. \(1\leq x\leq 4\). The closed interval ([1,4]) includes both endpoints. Therefore equality appears on both sides.
Step 3
Exam Tip
बंद अंतराल ([1,4]) दोनों सिरों को शामिल करता है। इसलिए दोनों ओर बराबरी वाला चिन्ह लगेगा।
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यदि \(x\in\mathbb{R}\) और (6<2x+4) है तो हल कौन सा है?
If \(x\in\mathbb{R}\) and (6<2x+4), what is the solution?
#reverse-order
#real-solution
#linear-inequality
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A (x>1)
B (x<1)
C (x>5)
D (x<5)
Explanation opens after your attempt
Step 1
Concept
From (6<2x+4), we get (2<2x), hence (x>1). Even when the inequality starts with a number, keep the order carefully.
Step 2
Why this answer is correct
The correct answer is A. (x>1). From (6<2x+4), we get (2<2x), hence (x>1). Even when the inequality starts with a number, keep the order carefully.
Step 3
Exam Tip
(6<2x+4) से (2<2x) और (x>1) मिलता है। असमता बाईं ओर संख्या से शुरू हो तो भी क्रम ध्यान से रखें।
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असमता \(\frac{3x-2}{2}<\frac{x+4}{3}\) का हल कौन सा है?
What is the solution of \(\frac{3x-2}{2}<\frac{x+4}{3}\)?
#fractional-linear-inequality
#lcm
#hard
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A (x<2)
B (x>2)
C (x< -2)
D (x> -2)
Explanation opens after your attempt
Step 1
Concept
Multiplying by (6) gives (9x-6<2x+8), so (7x<14). Removing positive denominators does not change the sign.
Step 2
Why this answer is correct
The correct answer is A. (x<2). Multiplying by (6) gives (9x-6<2x+8), so (7x<14). Removing positive denominators does not change the sign.
Step 3
Exam Tip
(6) से गुणा करने पर (9x-6<2x+8) और (7x<14) मिलता है। धनात्मक हर हटाने पर चिन्ह नहीं बदलता।
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कौन सा मान असमता \(-3\leq x<1\) का हल है?
Which value is a solution of \(-3\leq x<1\)?
#solution-check
#compound-inequality
#endpoint
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A (x=-3)
B (x=1)
C (x=-4)
D (x=2)
Explanation opens after your attempt
Step 1
Concept
(-3) is included because of \(\leq\), while (1) is not included because of (<). Check open and closed endpoints separately.
Step 2
Why this answer is correct
The correct answer is A. (x=-3). (-3) is included because of \(\leq\), while (1) is not included because of (<). Check open and closed endpoints separately.
Step 3
Exam Tip
(-3) शामिल है क्योंकि \(\leq\) है और (1) शामिल नहीं है क्योंकि (<) है। खुले और बंद सिरों को अलग-अलग जांचें।
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असमता \(10-5x\geq 0\) का हल किस अंतराल में है?
In which interval is the solution of \(10-5x\geq 0\)?
#interval-solution
#negative-coefficient
#closed-endpoint
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A (\(-\infty,2]\)
B \([2,\infty\))
C (\(-\infty,2\))
D (\(2,\infty\))
Explanation opens after your attempt
Correct Answer
A. (\(-\infty,2]\)
Step 1
Concept
From \(10-5x\geq 0\), \(-5x\geq -10\), so \(x\leq 2\). The boundary (2) is included because the sign was \(\geq\).
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,2]\). From \(10-5x\geq 0\), \(-5x\geq -10\), so \(x\leq 2\). The boundary (2) is included because the sign was \(\geq\).
Step 3
Exam Tip
\(10-5x\geq 0\) से \(-5x\geq -10\) और \(x\leq 2\) मिलता है। सीमा (2) शामिल है क्योंकि चिन्ह \(\geq\) था।
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यदि \(x\in\mathbb{Z}\) और \(1\leq 2x-1<9\) है तो हल समुच्चय कौन सा है?
If \(x\in\mathbb{Z}\) and \(1\leq 2x-1<9\), what is the solution set?
#integer-set
#compound-inequality
#solution
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A ({1,2,3,4})
B ({0,1,2,3,4})
C ({1,2,3,4,5})
D ({2,3,4})
Explanation opens after your attempt
Correct Answer
A. ({1,2,3,4})
Step 1
Concept
From \(1\leq 2x-1<9\), we get \(2\leq 2x<10\), hence \(1\leq x<5\). The integer solutions are (1) through (4).
Step 2
Why this answer is correct
The correct answer is A. ({1,2,3,4}). From \(1\leq 2x-1<9\), we get \(2\leq 2x<10\), hence \(1\leq x<5\). The integer solutions are (1) through (4).
Step 3
Exam Tip
\(1\leq 2x-1<9\) से \(2\leq 2x<10\) और \(1\leq x<5\) मिलता है। पूर्णांक हल (1) से (4) तक हैं।
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किस असमता का संख्या रेखा चित्र (0) पर बंद वृत्त और दाईं ओर छाया दिखाता है?
Which inequality has a number-line graph with a closed circle at (0) and shading to the right?
#number-line
#closed-circle
#representation
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A \(x\geq 0\)
B (x>0)
C \(x\leq 0\)
D (x<0)
Explanation opens after your attempt
Correct Answer
A. \(x\geq 0\)
Step 1
Concept
A closed circle includes (0), and right-side shading shows numbers greater than (0). Therefore \(x\geq 0\) is correct.
Step 2
Why this answer is correct
The correct answer is A. \(x\geq 0\). A closed circle includes (0), and right-side shading shows numbers greater than (0). Therefore \(x\geq 0\) is correct.
Step 3
Exam Tip
बंद वृत्त (0) को शामिल करता है और दाईं ओर छाया (0) से बड़ी संख्याओं को दिखाती है। इसलिए \(x\geq 0\) सही है।
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यदि \(2x-3\leq 5\) और (x+1>0) हैं तो संयुक्त हल कौन सा है?
If \(2x-3\leq 5\) and (x+1>0), what is the combined solution?
#combined-solution
#intersection
#linear-inequalities
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A \(-1<x\leq 4\)
B \(-1\leq x<4\)
C \(x\leq -1\)
D (x>4)
Explanation opens after your attempt
Correct Answer
A. \(-1<x\leq 4\)
Step 1
Concept
The first inequality gives \(x\leq 4\), and the second gives (x>-1). The combined solution is \(-1<x\leq 4\).
Step 2
Why this answer is correct
The correct answer is A. \(-1<x\leq 4\). The first inequality gives \(x\leq 4\), and the second gives (x>-1). The combined solution is \(-1<x\leq 4\).
Step 3
Exam Tip
पहली असमता से \(x\leq 4\) और दूसरी से (x>-1) मिलता है। संयुक्त हल \(-1<x\leq 4\) है।
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एक संख्या का (3) गुना (18) से कम नहीं है। यह कथन किस असमता से व्यक्त होगा?
Three times a number is not less than (18). Which inequality represents this statement?
#word-translation
#not-less-than
#concept
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A \(3x\geq 18\)
B (3x>18)
C \(3x\leq 18\)
D (3x<18)
Explanation opens after your attempt
Correct Answer
A. \(3x\geq 18\)
Step 1
Concept
Not less than means greater than or equal to (18). In word problems, match phrases correctly with inequality signs.
Step 2
Why this answer is correct
The correct answer is A. \(3x\geq 18\). Not less than means greater than or equal to (18). In word problems, match phrases correctly with inequality signs.
Step 3
Exam Tip
कम नहीं है का अर्थ (18) से बड़ा या बराबर है। भाषा आधारित प्रश्नों में शब्दों को असमता चिन्ह से सही मिलाएं।
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यदि \(x\in\mathbb{R}\) और (3-2x) धनात्मक है तो (x) का हल कौन सा है?
If \(x\in\mathbb{R}\) and (3-2x) is positive, what is the solution for (x)?
#positive-expression
#linear-inequality
#fraction
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A \(x<\frac{3}{2}\)
B \(x>\frac{3}{2}\)
C \(x<-\frac{3}{2}\)
D \(x>-\frac{3}{2}\)
Explanation opens after your attempt
Correct Answer
A. \(x<\frac{3}{2}\)
Step 1
Concept
Positive means (3-2x>0). This gives (-2x>-3), hence \(x<\frac{3}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(x<\frac{3}{2}\). Positive means (3-2x>0). This gives (-2x>-3), hence \(x<\frac{3}{2}\).
Step 3
Exam Tip
धनात्मक का अर्थ (3-2x>0) है। इससे (-2x>-3) और \(x<\frac{3}{2}\) मिलता है।
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यदि (5x+2) ऋणात्मक नहीं है तो (x) के लिए सही असमता कौन सी है?
If (5x+2) is non-negative, which inequality for (x) is correct?
#non-negative
#translation
#linear-inequality
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A \(x\geq -\frac{2}{5}\)
B \(x>-\frac{2}{5}\)
C \(x\leq -\frac{2}{5}\)
D \(x<-\frac{2}{5}\)
Explanation opens after your attempt
Correct Answer
A. \(x\geq -\frac{2}{5}\)
Step 1
Concept
Non-negative means \(5x+2\geq 0\). This gives \(x\geq -\frac{2}{5}\).
Step 2
Why this answer is correct
The correct answer is A. \(x\geq -\frac{2}{5}\). Non-negative means \(5x+2\geq 0\). This gives \(x\geq -\frac{2}{5}\).
Step 3
Exam Tip
ऋणात्मक नहीं है का अर्थ \(5x+2\geq 0\) है। इससे \(x\geq -\frac{2}{5}\) मिलता है।
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असमता (4-7x>18) का सही हल कौन सा है?
What is the correct solution of (4-7x>18)?
#linear-inequality
#negative-division
#exam-style
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A (x<-2)
B (x>-2)
C (x<2)
D (x>2)
Explanation opens after your attempt
Step 1
Concept
From (4-7x>18), we get (-7x>14), hence (x<-2). Do not forget to reverse the sign when dividing by negative (-7).
Step 2
Why this answer is correct
The correct answer is A. (x<-2). From (4-7x>18), we get (-7x>14), hence (x<-2). Do not forget to reverse the sign when dividing by negative (-7).
Step 3
Exam Tip
(4-7x>18) से (-7x>14) और (x<-2) मिलता है। ऋणात्मक (-7) से भाग देते समय चिन्ह पलटना न भूलें।
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