Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है • Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है • Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
C. (-6) पर खुला बिंदु और दाईं ओर रेखा/Open dot at (-6) and ray to the right
Step 1
Concept
The sign (>) does not include equality so the dot is open. In exams draw the ray to the right for greater values.
Step 2
Why this answer is correct
The correct answer is C. (-6) पर खुला बिंदु और दाईं ओर रेखा / Open dot at (-6) and ray to the right. The sign (>) does not include equality so the dot is open. In exams draw the ray to the right for greater values.
Step 3
Exam Tip
चिह्न (>) में बराबरी शामिल नहीं होती इसलिए खुला बिंदु लगेगा। परीक्षा में बड़े मानों के लिए रेखा दाईं ओर बनाएं।
The value (-2) is excluded but (3) is included so the integers are from (-1) to (3). In exams choose integer points according to open and closed endpoints.
Step 2
Why this answer is correct
The correct answer is D. ({-1,0,1,2,3}). The value (-2) is excluded but (3) is included so the integers are from (-1) to (3). In exams choose integer points according to open and closed endpoints.
Step 3
Exam Tip
(-2) शामिल नहीं है लेकिन (3) शामिल है इसलिए पूर्णांक (-1) से (3) तक होंगे। परीक्षा में खुले और बंद endpoints के अनुसार integer points चुनें।
The closed dot includes (-1) and the left direction shows smaller values. In exams use \(\le\) or \(\ge\) with a closed dot.
Step 2
Why this answer is correct
The correct answer is B. \(x\le -1\). The closed dot includes (-1) and the left direction shows smaller values. In exams use \(\le\) or \(\ge\) with a closed dot.
Step 3
Exam Tip
बंद बिंदु (-1) को शामिल करता है और बाईं दिशा छोटे मान दिखाती है। परीक्षा में बंद बिंदु के साथ \(\le\) या \(\ge\) का प्रयोग करें।
The closed dot includes (11) and the right direction shows greater values. In exams choose the equality sign with a closed dot.
Step 2
Why this answer is correct
The correct answer is C. \(x\ge 11\). The closed dot includes (11) and the right direction shows greater values. In exams choose the equality sign with a closed dot.
Step 3
Exam Tip
बंद बिंदु (11) को शामिल करता है और दाईं दिशा बड़े मान दिखाती है। परीक्षा में closed dot के साथ equality वाला चिन्ह चुनें।
B. \(\frac{1}{2}\) पर बंद और (4) पर खुला बिंदु/Closed dot at \(\frac{1}{2}\) and open dot at (4)
Step 1
Concept
The square bracket includes \(\frac{1}{2}\) and the round bracket excludes (4). In exams convert brackets directly into endpoints.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{1}{2}\) पर बंद और (4) पर खुला बिंदु / Closed dot at \(\frac{1}{2}\) and open dot at (4). The square bracket includes \(\frac{1}{2}\) and the round bracket excludes (4). In exams convert brackets directly into endpoints.
Step 3
Exam Tip
वर्ग कोष्ठक \(\frac{1}{2}\) को शामिल करता है और गोल कोष्ठक (4) को हटाता है। परीक्षा में brackets को सीधे endpoints में बदलें।
A. (3) पर खुला बिंदु और बाईं ओर रेखा/Open dot at (3) and ray to the left
Step 1
Concept
Solving gives (x<3) so (3) is not included. In exams first find the boundary value then decide open or closed dot.
Step 2
Why this answer is correct
The correct answer is A. (3) पर खुला बिंदु और बाईं ओर रेखा / Open dot at (3) and ray to the left. Solving gives (x<3) so (3) is not included. In exams first find the boundary value then decide open or closed dot.
Step 3
Exam Tip
हल करने पर (x<3) मिलता है इसलिए (3) शामिल नहीं है। परीक्षा में पहले सीमा मान निकालें फिर खुला या बंद बिंदु तय करें।
A. (4) पर खुला बिंदु और दाईं ओर रेखा/Open dot at (4) and ray right
Step 1
Concept
Solving gives (x>4) so (4) is not included. In exams first find the boundary and then decide the direction.
Step 2
Why this answer is correct
The correct answer is A. (4) पर खुला बिंदु और दाईं ओर रेखा / Open dot at (4) and ray right. Solving gives (x>4) so (4) is not included. In exams first find the boundary and then decide the direction.
Step 3
Exam Tip
हल करने पर (x>4) मिलता है इसलिए (4) शामिल नहीं है। परीक्षा में पहले boundary निकालें फिर direction तय करें।
The value (-6) is included and (-1) is excluded so \(-6\le x<-1\) is correct. In exams read negative endpoints in number line order.
Step 2
Why this answer is correct
The correct answer is B. \(-6\le x<-1\). The value (-6) is included and (-1) is excluded so \(-6\le x<-1\) is correct. In exams read negative endpoints in number line order.
Step 3
Exam Tip
(-6) शामिल है और (-1) शामिल नहीं है इसलिए \(-6\le x<-1\) सही है। परीक्षा में negative endpoints को भी number line order में पढ़ें।
Multiplying by (5) and subtracting (3) gives \(7\le x<12\). In exams positive multiplication does not reverse the inequality sign.
Step 2
Why this answer is correct
The correct answer is A. \(7\le x<12\). Multiplying by (5) and subtracting (3) gives \(7\le x<12\). In exams positive multiplication does not reverse the inequality sign.
Step 3
Exam Tip
(5) से गुणा करके और (3) घटाकर \(7\le x<12\) मिलता है। परीक्षा में positive multiplication से inequality sign नहीं बदलता।
The closed dot includes (8) and the right direction shows greater values. In exams choose (>) or \(\ge\) for a right ray.
Step 2
Why this answer is correct
The correct answer is B. \(x\ge 8\). The closed dot includes (8) and the right direction shows greater values. In exams choose (>) or \(\ge\) for a right ray.
Step 3
Exam Tip
बंद बिंदु (8) को शामिल करता है और दाईं दिशा बड़े मान दिखाती है। परीक्षा में दाईं रेखा के लिए (>) या \(\ge\) चुनें।
For or take the union of both separate parts. In exams keep the \(\le\) endpoint closed and the (>) endpoint open.
Step 2
Why this answer is correct
The correct answer is C. ((-\infty,-5]\cup\(2,\infty\)). For or take the union of both separate parts. In exams keep the \(\le\) endpoint closed and the (>) endpoint open.
Step 3
Exam Tip
or में दोनों अलग-अलग भागों का union लिया जाता है। परीक्षा में \(\le\) वाले endpoint को बंद और (>) वाले endpoint को खुला रखें।
The value (-5) is included and (2) is excluded so ([-5,2)) is correct. In exams use a square bracket at the equality endpoint.
Step 2
Why this answer is correct
The correct answer is A. ([-5,2)). The value (-5) is included and (2) is excluded so ([-5,2)) is correct. In exams use a square bracket at the equality endpoint.
Step 3
Exam Tip
(-5) शामिल है और (2) शामिल नहीं है इसलिए ([-5,2)) सही है। परीक्षा में बराबरी वाले सिरे पर वर्ग कोष्ठक लगाएं।
Dividing by (-3) reverses both signs and gives \(-2\le x<5\). In exams fix the endpoint order after negative division.
Step 2
Why this answer is correct
The correct answer is A. \(-2\le x<5\). Dividing by (-3) reverses both signs and gives \(-2\le x<5\). In exams fix the endpoint order after negative division.
Step 3
Exam Tip
(-3) से भाग देने पर दोनों चिन्ह पलटते हैं और \(-2\le x<5\) मिलता है। परीक्षा में negative division के बाद endpoints का क्रम सही करें।
D. \(\frac{3}{2}\) पर बंद बिंदु और बाईं ओर रेखा/Closed dot at \(\frac{3}{2}\) and ray left
Step 1
Concept
The sign \(\le\) includes equality and smaller values lie to the left. In exams place a fractional boundary like any ordinary number.
Step 2
Why this answer is correct
The correct answer is D. \(\frac{3}{2}\) पर बंद बिंदु और बाईं ओर रेखा / Closed dot at \(\frac{3}{2}\) and ray left. The sign \(\le\) includes equality and smaller values lie to the left. In exams place a fractional boundary like any ordinary number.
Step 3
Exam Tip
\(\le\) में बराबरी शामिल है और छोटे मान बाईं ओर होते हैं। परीक्षा में भिन्न सीमा मान को भी सामान्य संख्या की तरह रखें।
The open dot excludes (-3) and the closed dot includes (6). In exams write the shaded region as a compound inequality.
Step 2
Why this answer is correct
The correct answer is C. \(-3<x\le 6\). The open dot excludes (-3) and the closed dot includes (6). In exams write the shaded region as a compound inequality.
Step 3
Exam Tip
खुला बिंदु (-3) को हटाता है और बंद बिंदु (6) को शामिल करता है। परीक्षा में छायांकित भाग को बीच की असमानता में लिखें।
B. \(\frac{5}{3}\) पर बंद बिंदु और बाईं ओर रेखा/Closed dot at \(\frac{5}{3}\) and ray left
Step 1
Concept
The value \(\frac{5}{3}\) is included and all smaller values are taken. In exams understand (\(-\infty,a]\) as \(x\le a\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{5}{3}\) पर बंद बिंदु और बाईं ओर रेखा / Closed dot at \(\frac{5}{3}\) and ray left. The value \(\frac{5}{3}\) is included and all smaller values are taken. In exams understand (\(-\infty,a]\) as \(x\le a\).
Step 3
Exam Tip
\(\frac{5}{3}\) शामिल है और उससे छोटे सभी मान लिए जाते हैं। परीक्षा में (\(-\infty,a]\) को \(x\le a\) समझें।
Solving gives \(-2x\ge -4\) and dividing by a negative changes the sign to \(x\le 2\). In exams reverse the inequality sign when dividing by a negative.
Step 2
Why this answer is correct
The correct answer is B. \(x\le 2\). Solving gives \(-2x\ge -4\) and dividing by a negative changes the sign to \(x\le 2\). In exams reverse the inequality sign when dividing by a negative.
Step 3
Exam Tip
हल करने पर \(-2x\ge -4\) और ऋणात्मक से भाग देने पर चिह्न बदलकर \(x\le 2\) होता है। परीक्षा में ऋणात्मक से भाग देने पर असमानता का चिन्ह पलटें।
A. (9) पर बंद बिंदु और बाईं ओर रेखा/Closed dot at (9) and ray to the left
Step 1
Concept
The interval (\(-\infty,9]\) includes (9) and all smaller values. In exams the ray toward \(-\infty\) goes left.
Step 2
Why this answer is correct
The correct answer is A. (9) पर बंद बिंदु और बाईं ओर रेखा / Closed dot at (9) and ray to the left. The interval (\(-\infty,9]\) includes (9) and all smaller values. In exams the ray toward \(-\infty\) goes left.
Step 3
Exam Tip
(\(-\infty,9]\) में (9) शामिल है और सभी छोटे मान लिए जाते हैं। परीक्षा में \(-\infty\) की ओर रेखा बाईं दिशा में होती है।
B. (4) पर खुला बिंदु और बाकी पूरी रेखा/Open dot at (4) and the rest of the whole line
Step 1
Concept
The inequality \(x\ne 4\) includes all real numbers except (4). In exams show the removed point as an open dot.
Step 2
Why this answer is correct
The correct answer is B. (4) पर खुला बिंदु और बाकी पूरी रेखा / Open dot at (4) and the rest of the whole line. The inequality \(x\ne 4\) includes all real numbers except (4). In exams show the removed point as an open dot.
Step 3
Exam Tip
\(x\ne 4\) में (4) को छोड़कर सभी वास्तविक संख्याएँ आती हैं। परीक्षा में हटाए गए बिंदु को खुला दिखाएं।
C. (-7) शामिल नहीं है और (-2) शामिल है/(-7) is excluded and (-2) is included
Step 1
Concept
The round bracket excludes (-7) and the square bracket includes (-2). In exams the endpoint rule stays the same for negative numbers.
Step 2
Why this answer is correct
The correct answer is C. (-7) शामिल नहीं है और (-2) शामिल है / (-7) is excluded and (-2) is included. The round bracket excludes (-7) and the square bracket includes (-2). In exams the endpoint rule stays the same for negative numbers.
Step 3
Exam Tip
गोल कोष्ठक (-7) को हटाता है और वर्ग कोष्ठक (-2) को शामिल करता है। परीक्षा में ऋणात्मक संख्याओं में भी सिरों का नियम वही रहता है।
The open dot excludes (1) and the right direction shows greater values. In exams infinity endpoints always use round brackets.
Step 2
Why this answer is correct
The correct answer is C. (\(1,\infty\)). The open dot excludes (1) and the right direction shows greater values. In exams infinity endpoints always use round brackets.
Step 3
Exam Tip
खुला बिंदु (1) को हटाता है और दाईं दिशा बड़े मानों को दिखाती है। परीक्षा में \(\infty\) वाले सिरे पर हमेशा गोल कोष्ठक होता है।
C. (-1) और (3) दोनों पर बंद बिंदु/Closed dots at both (-1) and (3)
Step 1
Concept
The interval ([-1,3]) has square brackets at both ends so both values are included. In exams draw closed dots for both endpoints of a closed interval.
Step 2
Why this answer is correct
The correct answer is C. (-1) और (3) दोनों पर बंद बिंदु / Closed dots at both (-1) and (3). The interval ([-1,3]) has square brackets at both ends so both values are included. In exams draw closed dots for both endpoints of a closed interval.
Step 3
Exam Tip
([-1,3]) में दोनों सिरों पर वर्ग कोष्ठक हैं इसलिए दोनों मान शामिल हैं। परीक्षा में closed interval के दोनों बिंदु बंद बनाएं।
Both closed dots show that (-8) and (2) are included. In exams connect closed dots with \(\le\) signs on both sides.
Step 2
Why this answer is correct
The correct answer is B. \(-8\le x\le 2\). Both closed dots show that (-8) and (2) are included. In exams connect closed dots with \(\le\) signs on both sides.
Step 3
Exam Tip
दोनों बंद बिंदु बताते हैं कि (-8) और (2) शामिल हैं। परीक्षा में closed dots को \(\le\) वाले दोनों चिह्नों से जोड़ें।
A. \(-\frac{5}{2}\) पर बंद बिंदु और दाईं ओर रेखा/Closed dot at \(-\frac{5}{2}\) and ray right
Step 1
Concept
Because of \(\ge\) the boundary value is included and greater values are to the right. In exams place the negative fraction correctly on the number line.
Step 2
Why this answer is correct
The correct answer is A. \(-\frac{5}{2}\) पर बंद बिंदु और दाईं ओर रेखा / Closed dot at \(-\frac{5}{2}\) and ray right. Because of \(\ge\) the boundary value is included and greater values are to the right. In exams place the negative fraction correctly on the number line.
Step 3
Exam Tip
\(\ge\) के कारण सीमा मान शामिल है और बड़े मान दाईं ओर हैं। परीक्षा में ऋणात्मक भिन्न को संख्या रेखा पर सही स्थान पर रखें।
C. जब असमानता में (<) या (>) हो/When the inequality has (<) or (>)
Step 1
Concept
In a strict inequality the boundary value is not included so the endpoint is open. In exams connect (<) and (>) with open dots.
Step 2
Why this answer is correct
The correct answer is C. जब असमानता में (<) या (>) हो / When the inequality has (<) or (>). In a strict inequality the boundary value is not included so the endpoint is open. In exams connect (<) and (>) with open dots.
Step 3
Exam Tip
सख्त असमानता में सीमा मान शामिल नहीं होता इसलिए endpoint खुला होता है। परीक्षा में (<) और (>) को open dot से जोड़ें।
The value (-4) is excluded and all smaller values are included so (x<-4). In exams read an interval with \(-\infty\) as a left ray.
Step 2
Why this answer is correct
The correct answer is B. (x<-4). The value (-4) is excluded and all smaller values are included so (x<-4). In exams read an interval with \(-\infty\) as a left ray.
Step 3
Exam Tip
(-4) शामिल नहीं है और सभी छोटे मान शामिल हैं इसलिए (x<-4) होगा। परीक्षा में \(-\infty\) वाले अंतराल को बाईं रेखा समझें।
The open dot excludes (6) and right shading shows values greater than (6). In exams never use a square bracket with \(\infty\).
Step 2
Why this answer is correct
The correct answer is C. (\(6,\infty\)). The open dot excludes (6) and right shading shows values greater than (6). In exams never use a square bracket with \(\infty\).
Step 3
Exam Tip
खुला बिंदु (6) को हटाता है और दाईं छाया (6) से बड़े मान दिखाती है। परीक्षा में \(\infty\) पर कभी वर्ग कोष्ठक न लगाएं।
B. \(x\ge 2\) के रूप में बंद बिंदु और दाईं रेखा/As \(x\ge 2\) with closed dot and right ray
Step 1
Concept
The solution gives \(-4x\le -8\) and dividing by a negative gives \(x\ge 2\). In exams forgetting sign reversal is the most common error.
Step 2
Why this answer is correct
The correct answer is B. \(x\ge 2\) के रूप में बंद बिंदु और दाईं रेखा / As \(x\ge 2\) with closed dot and right ray. The solution gives \(-4x\le -8\) and dividing by a negative gives \(x\ge 2\). In exams forgetting sign reversal is the most common error.
Step 3
Exam Tip
हल में \(-4x\le -8\) आता है और ऋणात्मक से भाग देने पर \(x\ge 2\) मिलता है। परीक्षा में sign reversal भूलना सबसे सामान्य गलती है।
A. (3) पर खुला और (8) पर बंद बिंदु/Open dot at (3) and closed dot at (8)
Step 1
Concept
In ((3,8]) the value (3) is excluded and (8) is included. In exams convert interval notation directly into endpoint dots.
Step 2
Why this answer is correct
The correct answer is A. (3) पर खुला और (8) पर बंद बिंदु / Open dot at (3) and closed dot at (8). In ((3,8]) the value (3) is excluded and (8) is included. In exams convert interval notation directly into endpoint dots.
Step 3
Exam Tip
((3,8]) में (3) हटता है और (8) शामिल होता है। परीक्षा में interval notation को सीधे endpoint dots में बदलें।
Multiplying the whole inequality by (2) gives \(2\le x<8\). In exams the sign does not change when multiplying by a positive number.
Step 2
Why this answer is correct
The correct answer is B. \(2\le x<8\). Multiplying the whole inequality by (2) gives \(2\le x<8\). In exams the sign does not change when multiplying by a positive number.
Step 3
Exam Tip
पूरी असमानता को (2) से गुणा करने पर \(2\le x<8\) मिलता है। परीक्षा में धनात्मक से गुणा करने पर चिन्ह नहीं बदलता।
The value (-2) is included and (5) is excluded so ([-2,5)) is correct. In exams read the left and right endpoints separately.
Step 2
Why this answer is correct
The correct answer is C. ([-2,5)). The value (-2) is included and (5) is excluded so ([-2,5)) is correct. In exams read the left and right endpoints separately.
Step 3
Exam Tip
(-2) शामिल है और (5) शामिल नहीं है इसलिए ([-2,5)) सही है। परीक्षा में left endpoint और right endpoint अलग-अलग पढ़ें।
D. (0) पर बंद बिंदु और दाईं ओर रेखा/Closed dot at (0) and ray right
Step 1
Concept
In \(x\ge 0\) the value (0) is included and greater values are to the right. In exams treat (0) as a boundary value too.
Step 2
Why this answer is correct
The correct answer is D. (0) पर बंद बिंदु और दाईं ओर रेखा / Closed dot at (0) and ray right. In \(x\ge 0\) the value (0) is included and greater values are to the right. In exams treat (0) as a boundary value too.
Step 3
Exam Tip
\(x\ge 0\) में (0) शामिल है और उससे बड़े मान दाईं ओर हैं। परीक्षा में (0) को भी सीमा मान की तरह शामिल करें।
This inequality is satisfied only by (x=2) so one closed dot is drawn. In exams treat a closed interval with equal endpoints as a single point.
Step 2
Why this answer is correct
The correct answer is B. (2) पर केवल बंद बिंदु / Only a closed dot at (2). This inequality is satisfied only by (x=2) so one closed dot is drawn. In exams treat a closed interval with equal endpoints as a single point.
Step 3
Exam Tip
यह असमानता केवल (x=2) को संतुष्ट करती है इसलिए एक बंद बिंदु बनेगा। परीक्षा में समान दोनों सिरों वाले closed interval को single point मानें।
In ((4,4)) both endpoints are open and no number remains between them. In exams an interval with equal open endpoints is empty.
Step 2
Why this answer is correct
The correct answer is C. कोई वास्तविक संख्या नहीं / No real number. In ((4,4)) both endpoints are open and no number remains between them. In exams an interval with equal open endpoints is empty.
Step 3
Exam Tip
((4,4)) में दोनों सिर खुले हैं और बीच में कोई संख्या नहीं बचती। परीक्षा में समान open endpoints वाला interval empty होता है।
There is equality at (-6) so use a square bracket and no equality at (-1) so use a round bracket. In exams practice converting signs into brackets.
Step 2
Why this answer is correct
The correct answer is B. ([-6,-1)). There is equality at (-6) so use a square bracket and no equality at (-1) so use a round bracket. In exams practice converting signs into brackets.
Step 3
Exam Tip
(-6) पर बराबरी है इसलिए वर्ग कोष्ठक और (-1) पर बराबरी नहीं है इसलिए गोल कोष्ठक होगा। परीक्षा में signs को brackets में बदलना अभ्यास करें।
The open dot excludes the boundary and the left direction shows smaller values. In exams pay special attention to the direction for a negative fraction.
Step 2
Why this answer is correct
The correct answer is C. \(x<-\frac{1}{3}\). The open dot excludes the boundary and the left direction shows smaller values. In exams pay special attention to the direction for a negative fraction.
Step 3
Exam Tip
खुला बिंदु सीमा को हटाता है और बाईं दिशा छोटे मान दिखाती है। परीक्षा में negative fraction की दिशा पर खास ध्यान दें।
B. (0) पर बंद बिंदु और बाईं ओर रेखा/Closed dot at (0) and ray left
Step 1
Concept
The interval (\(-\infty,0]\) includes (0) and all smaller values. In exams remember (\(-\infty,0]\) as \(x\le 0\).
Step 2
Why this answer is correct
The correct answer is B. (0) पर बंद बिंदु और बाईं ओर रेखा / Closed dot at (0) and ray left. The interval (\(-\infty,0]\) includes (0) and all smaller values. In exams remember (\(-\infty,0]\) as \(x\le 0\).
Step 3
Exam Tip
(\(-\infty,0]\) में (0) शामिल है और उससे छोटे मान शामिल हैं। परीक्षा में (\(-\infty,0]\) को \(x\le 0\) से जोड़कर याद रखें।
Simplifying gives \(x\le 6\) so there is a closed dot at (6) and a left ray. In exams bring variable terms to one side before representing.
Step 2
Why this answer is correct
The correct answer is C. \(x\le 6\). Simplifying gives \(x\le 6\) so there is a closed dot at (6) and a left ray. In exams bring variable terms to one side before representing.
Step 3
Exam Tip
सरल करने पर \(x\le 6\) मिलता है इसलिए (6) पर बंद बिंदु और बाईं रेखा होगी। परीक्षा में चर पदों को एक तरफ लाकर हल करें।
For or take the union of both separate regions. In exams keep a closed endpoint for \(\le -2\) and an open endpoint for (>3).
Step 2
Why this answer is correct
The correct answer is B. ((-\infty,-2]\cup\(3,\infty\)). For or take the union of both separate regions. In exams keep a closed endpoint for \(\le -2\) and an open endpoint for (>3).
Step 3
Exam Tip
or में दोनों अलग-अलग भागों का union लिया जाता है। परीक्षा में \(\le -2\) पर बंद और (>3) पर खुला endpoint रखें।
B. \(x\ge 3\) के रूप में दाईं रेखा/As \(x\ge 3\) with right ray
Step 1
Concept
The inequality \(3\le x\) can also be written as \(x\ge 3\). In exams writing the variable on the left makes the direction easier to read.
Step 2
Why this answer is correct
The correct answer is B. \(x\ge 3\) के रूप में दाईं रेखा / As \(x\ge 3\) with right ray. The inequality \(3\le x\) can also be written as \(x\ge 3\). In exams writing the variable on the left makes the direction easier to read.
Step 3
Exam Tip
\(3\le x\) को \(x\ge 3\) भी लिखा जा सकता है। परीक्षा में चर को बाईं ओर लिखकर दिशा समझना आसान होता है।
B. (0) बंद और \(\frac{9}{4}\) खुला/(0) closed and \(\frac{9}{4}\) open
Step 1
Concept
There is equality at (0) so it is closed and no equality at \(\frac{9}{4}\) so it is open. In exams draw mixed endpoints carefully.
Step 2
Why this answer is correct
The correct answer is B. (0) बंद और \(\frac{9}{4}\) खुला / (0) closed and \(\frac{9}{4}\) open. There is equality at (0) so it is closed and no equality at \(\frac{9}{4}\) so it is open. In exams draw mixed endpoints carefully.
Step 3
Exam Tip
(0) पर बराबरी है इसलिए बंद बिंदु और \(\frac{9}{4}\) पर बराबरी नहीं है इसलिए खुला बिंदु होगा। परीक्षा में mixed endpoints को ध्यान से बनाएं।
B. (7) पर बंद बिंदु और दाईं रेखा/Closed dot at (7) and ray right
Step 1
Concept
Solving gives \(x-1\ge 6\) and \(x\ge 7\). In exams keep the endpoint closed for \(\ge\).
Step 2
Why this answer is correct
The correct answer is B. (7) पर बंद बिंदु और दाईं रेखा / Closed dot at (7) and ray right. Solving gives \(x-1\ge 6\) and \(x\ge 7\). In exams keep the endpoint closed for \(\ge\).
Step 3
Exam Tip
हल करने पर \(x-1\ge 6\) और \(x\ge 7\) मिलता है। परीक्षा में \(\ge\) होने पर endpoint बंद रखें।
The whole real line is written as (\(-\infty,\infty\)) and square brackets are not used with infinity. In exams do not treat infinity as an includable endpoint.
Step 2
Why this answer is correct
The correct answer is A. (\(-\infty,\infty\)). The whole real line is written as (\(-\infty,\infty\)) and square brackets are not used with infinity. In exams do not treat infinity as an includable endpoint.
Step 3
Exam Tip
पूरी वास्तविक रेखा (\(-\infty,\infty\)) से लिखी जाती है और infinity पर वर्ग कोष्ठक नहीं लगते। परीक्षा में infinity को endpoint मानकर शामिल न करें।
The open dot excludes (-9) and the right ray takes values greater than it. In exams write (>) as (\(a,\infty\)).
Step 2
Why this answer is correct
The correct answer is C. (\(-9,\infty\)). The open dot excludes (-9) and the right ray takes values greater than it. In exams write (>) as (\(a,\infty\)).
Step 3
Exam Tip
खुला बिंदु (-9) को हटाता है और दाईं रेखा उससे बड़े मान लेती है। परीक्षा में (>) को (\(a,\infty\)) के रूप में लिखें।
The two parts are separate and both (1) and (4) are included. In exams show both rays separately for a union.
Step 2
Why this answer is correct
The correct answer is C. (\(-\infty,1]\cup[4,\infty\)). The two parts are separate and both (1) and (4) are included. In exams show both rays separately for a union.
Step 3
Exam Tip
दोनों भाग अलग हैं और (1) तथा (4) दोनों शामिल हैं। परीक्षा में union में दोनों rays को अलग-अलग दिखाएं।
Dividing by (-3) reverses both inequality signs and gives \(-2<x\le 4\). In exams keep the order correct after negative division.
Step 2
Why this answer is correct
The correct answer is B. \(-2<x\le 4\). Dividing by (-3) reverses both inequality signs and gives \(-2<x\le 4\). In exams keep the order correct after negative division.
Step 3
Exam Tip
(-3) से भाग देने पर दोनों असमानता चिन्ह पलटते हैं और \(-2<x\le 4\) मिलता है। परीक्षा में negative division के बाद क्रम को सही रखें।
