Substituting ((0,2)) gives \(2\leq 1\), which is false. Decide the region only after comparison in exams.
Step 2
Why this answer is correct
The correct answer is C. हल क्षेत्र में नहीं / Not in the solution region. Substituting ((0,2)) gives \(2\leq 1\), which is false. Decide the region only after comparison in exams.
Step 3
Exam Tip
((0,2)) रखने पर \(2\leq 1\) असत्य है। परीक्षा में तुलना के बाद ही क्षेत्र तय करें।
From \(x-2y\leq 0\), we get \(x\leq 2y\), so \(y\geq \frac{x}{2}\). Remember sign changes when handling negative coefficients in exams.
Step 2
Why this answer is correct
The correct answer is B. \(y\geq \frac{x}{2}\). From \(x-2y\leq 0\), we get \(x\leq 2y\), so \(y\geq \frac{x}{2}\). Remember sign changes when handling negative coefficients in exams.
Step 3
Exam Tip
\(x-2y\leq 0\) से \(x\leq 2y\), इसलिए \(y\geq \frac{x}{2}\) मिलता है। परीक्षा में ऋणात्मक गुणांक से भाग करते समय संकेत बदलना याद रखें।
Substituting ((3,0)) gives \(9\leq 6\), which is false, so it is not in the solution region. In exams, find the option that makes the inequality false.
Step 2
Why this answer is correct
The correct answer is C. ((3,0)). Substituting ((3,0)) gives \(9\leq 6\), which is false, so it is not in the solution region. In exams, find the option that makes the inequality false.
Step 3
Exam Tip
((3,0)) रखने पर \(9\leq 6\) असत्य है, इसलिए यह हल क्षेत्र में नहीं है। परीक्षा में सभी दिए बिंदुओं में केवल असत्य मान खोजें।
For the boundary line (x+3y=6), (y=0) gives (x=6) and (x=0) gives (y=2). Find both intercepts separately in exams.
Step 2
Why this answer is correct
The correct answer is B. ((6,0)) और ((0,2)) / ((6,0)) and ((0,2)). For the boundary line (x+3y=6), (y=0) gives (x=6) and (x=0) gives (y=2). Find both intercepts separately in exams.
Step 3
Exam Tip
सीमा रेखा (x+3y=6) में (y=0) पर (x=6) और (x=0) पर (y=2) मिलता है। परीक्षा में दोनों प्रतिच्छेद अलग-अलग निकालें।
B. सीमा रेखा पर और हल में/On the boundary line and in the solution
Step 1
Concept
Substituting ((2,0)) gives \(4\geq 4\), true with equality. Boundary points are included when the symbol is \(\geq\).
Step 2
Why this answer is correct
The correct answer is B. सीमा रेखा पर और हल में / On the boundary line and in the solution. Substituting ((2,0)) gives \(4\geq 4\), true with equality. Boundary points are included when the symbol is \(\geq\).
Step 3
Exam Tip
((2,0)) रखने पर \(4\geq 4\) सत्य है और बराबरी भी है। परीक्षा में \(\geq\) होने पर सीमा बिंदु शामिल होते हैं।
B. यह हल क्षेत्र में है/It is in the solution region
Step 1
Concept
Substituting ((3,0)) gives (3>2), which is true, so it lies in the solution region. In exams, check the candidate point directly.
Step 2
Why this answer is correct
The correct answer is B. यह हल क्षेत्र में है / It is in the solution region. Substituting ((3,0)) gives (3>2), which is true, so it lies in the solution region. In exams, check the candidate point directly.
Step 3
Exam Tip
((3,0)) रखने पर (3>2) सत्य है, इसलिए यह हल क्षेत्र में है। परीक्षा में एक-एक विकल्प को रखने की जरूरत नहीं, सही बिंदु जांचें।
A. निचला अर्ध-तल, (x)-अक्ष सहित/Lower half-plane including the (x)-axis
Step 1
Concept
In \(y\leq 0\), (y) is zero or negative, so the (x)-axis and the lower side are included. Remember (y=0) is the (x)-axis in exams.
Step 2
Why this answer is correct
The correct answer is A. निचला अर्ध-तल, (x)-अक्ष सहित / Lower half-plane including the (x)-axis. In \(y\leq 0\), (y) is zero or negative, so the (x)-axis and the lower side are included. Remember (y=0) is the (x)-axis in exams.
Step 3
Exam Tip
\(y\leq 0\) में (y) शून्य या ऋणात्मक है, इसलिए (x)-अक्ष और नीचे का भाग शामिल है। परीक्षा में (y=0) को (x)-अक्ष समझें।
C. दायां अर्ध-तल, (y)-अक्ष सहित/Right half-plane including the (y)-axis
Step 1
Concept
In \(x\geq 0\), (x) is non-negative, so the (y)-axis and the right side are included. Remember (x=0) is the (y)-axis in exams.
Step 2
Why this answer is correct
The correct answer is C. दायां अर्ध-तल, (y)-अक्ष सहित / Right half-plane including the (y)-axis. In \(x\geq 0\), (x) is non-negative, so the (y)-axis and the right side are included. Remember (x=0) is the (y)-axis in exams.
Step 3
Exam Tip
\(x\geq 0\) में (x) गैर-ऋणात्मक है, इसलिए (y)-अक्ष और दायां भाग शामिल हैं। परीक्षा में (x=0) को (y)-अक्ष समझें।
B. रेखा (y=-2) के ऊपर, रेखा छोड़कर/Above the line (y=-2), excluding the line
Step 1
Concept
In (y> -2), (y)-values are greater than (-2), so shade above the line. Do not include the boundary line for (>) in exams.
Step 2
Why this answer is correct
The correct answer is B. रेखा (y=-2) के ऊपर, रेखा छोड़कर / Above the line (y=-2), excluding the line. In (y> -2), (y)-values are greater than (-2), so shade above the line. Do not include the boundary line for (>) in exams.
Step 3
Exam Tip
(y> -2) में (y) के मान (-2) से बड़े हैं, इसलिए रेखा के ऊपर छायांकन होगा। परीक्षा में (>) के लिए सीमा रेखा शामिल न करें।
Substituting ((1,1)) gives \(1\leq 2\), which is true, so the point is in the solution region. Calculate the right side carefully in exams.
Step 2
Why this answer is correct
The correct answer is C. हल क्षेत्र में / In the solution region. Substituting ((1,1)) gives \(1\leq 2\), which is true, so the point is in the solution region. Calculate the right side carefully in exams.
Step 3
Exam Tip
((1,1)) रखने पर \(1\leq 2\) सत्य है, इसलिए बिंदु हल क्षेत्र में है। परीक्षा में दाईं ओर का मान सावधानी से निकालें।
The boundary line is (2x+3y=12), giving (x=6) when (y=0) and (y=4) when (x=0). The intercept method draws the line quickly in exams.
Step 2
Why this answer is correct
The correct answer is A. ((6,0)) और ((0,4)) / ((6,0)) and ((0,4)). The boundary line is (2x+3y=12), giving (x=6) when (y=0) and (y=4) when (x=0). The intercept method draws the line quickly in exams.
Step 3
Exam Tip
सीमा रेखा (2x+3y=12) है, जिसमें (y=0) पर (x=6) और (x=0) पर (y=4) मिलता है। परीक्षा में प्रतिच्छेद विधि से रेखा जल्दी बनती है।
B. सीमा रेखा के विपरीत मूल बिंदु से दूर वाला भाग/Side opposite to the origin
Step 1
Concept
Substituting ((0,0)) gives \(0\geq 7\), which is false, so the solution is the side opposite the origin. Shade the other side when the test is false in exams.
Step 2
Why this answer is correct
The correct answer is B. सीमा रेखा के विपरीत मूल बिंदु से दूर वाला भाग / Side opposite to the origin. Substituting ((0,0)) gives \(0\geq 7\), which is false, so the solution is the side opposite the origin. Shade the other side when the test is false in exams.
Step 3
Exam Tip
((0,0)) रखने पर \(0\geq 7\) असत्य है, इसलिए मूल बिंदु के विपरीत भाग हल है। परीक्षा में असत्य परीक्षण पर दूसरी ओर छायांकन करें।
A. मूल बिंदु वाला भाग हल क्षेत्र है/The side containing the origin is the solution region
Step 1
Concept
Substituting ((0,0)) gives \(0\geq -2\), which is true, so shade the origin side. If the test point is true, take that side in exams.
Step 2
Why this answer is correct
The correct answer is A. मूल बिंदु वाला भाग हल क्षेत्र है / The side containing the origin is the solution region. Substituting ((0,0)) gives \(0\geq -2\), which is true, so shade the origin side. If the test point is true, take that side in exams.
Step 3
Exam Tip
((0,0)) रखने पर \(0\geq -2\) सत्य है, इसलिए मूल बिंदु वाला भाग छायांकित होगा। परीक्षा में परीक्षण बिंदु सत्य हो तो वही भाग लें।
B. रेखा (x=4) के बाईं ओर ठोस रेखा सहित/Left of (x=4) including solid line
Step 1
Concept
In \(x\leq 4\), (x)-values are (4) or less, so the region is to the left. In exams, read \(x\leq a\) as the left side.
Step 2
Why this answer is correct
The correct answer is B. रेखा (x=4) के बाईं ओर ठोस रेखा सहित / Left of (x=4) including solid line. In \(x\leq 4\), (x)-values are (4) or less, so the region is to the left. In exams, read \(x\leq a\) as the left side.
Step 3
Exam Tip
\(x\leq 4\) में (x) के मान (4) या कम हैं, इसलिए बाईं ओर क्षेत्र होगा। परीक्षा में \(x\leq a\) को बाईं ओर समझें।
B. यह हल क्षेत्र में नहीं है/It is not in the solution region
Step 1
Concept
Substituting ((1,3)) gives (3<2), which is false. In exams, directly substitute the point in the inequality.
Step 2
Why this answer is correct
The correct answer is B. यह हल क्षेत्र में नहीं है / It is not in the solution region. Substituting ((1,3)) gives (3<2), which is false. In exams, directly substitute the point in the inequality.
Step 3
Exam Tip
((1,3)) रखने पर (3<2) असत्य है। परीक्षा में बिंदु को असमानता में सीधे रखें।
A. हाँ, क्योंकि \(3\leq 3\)/Yes, because \(3\leq 3\)
Step 1
Concept
Substituting ((2,1)) gives (2+1=3), and \(3\leq 3\) is true. Boundary points are included for \(\leq\) in exams.
Step 2
Why this answer is correct
The correct answer is A. हाँ, क्योंकि \(3\leq 3\) / Yes, because \(3\leq 3\). Substituting ((2,1)) gives (2+1=3), and \(3\leq 3\) is true. Boundary points are included for \(\leq\) in exams.
Step 3
Exam Tip
((2,1)) रखने पर (2+1=3) और \(3\leq 3\) सत्य है। परीक्षा में सीमा पर स्थित बिंदु भी \(\leq\) में शामिल होते हैं।
Equality is not included in (x-y<2), so the boundary line (x-y=2) is dashed. In exams, do not include the line for (<).
Step 2
Why this answer is correct
The correct answer is B. टूटी हुई रेखा (x-y=2) / Dashed line (x-y=2). Equality is not included in (x-y<2), so the boundary line (x-y=2) is dashed. In exams, do not include the line for (<).
Step 3
Exam Tip
(x-y<2) में बराबरी शामिल नहीं है, इसलिए सीमा रेखा (x-y=2) टूटी होगी। परीक्षा में (<) देखकर रेखा को शामिल न करें।
A. ((0,0)) हल क्षेत्र में है/((0,0)) is in the solution region
Step 1
Concept
Substituting ((0,0)) gives \(0\leq 9\), so the side containing the origin is selected. Use a test point to decide the shading direction in exams.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)) हल क्षेत्र में है / ((0,0)) is in the solution region. Substituting ((0,0)) gives \(0\leq 9\), so the side containing the origin is selected. Use a test point to decide the shading direction in exams.
Step 3
Exam Tip
((0,0)) रखने पर \(0\leq 9\) सत्य है, इसलिए मूल बिंदु वाला भाग चुना जाएगा। परीक्षा में परीक्षण बिंदु से छायांकन की दिशा तय करें।
C. हाँ, क्योंकि \(\geq\) है/Yes, because it has \(\geq\)
Step 1
Concept
The symbol \(\geq\) includes equality, so the boundary line is part of the solution. Draw a solid line for \(\leq\) and \(\geq\) in exams.
Step 2
Why this answer is correct
The correct answer is C. हाँ, क्योंकि \(\geq\) है / Yes, because it has \(\geq\). The symbol \(\geq\) includes equality, so the boundary line is part of the solution. Draw a solid line for \(\leq\) and \(\geq\) in exams.
Step 3
Exam Tip
\(\geq\) में बराबरी शामिल होती है, इसलिए सीमा रेखा भी हल में आती है। परीक्षा में \(\leq\) और \(\geq\) के लिए ठोस रेखा बनाएं।
Substituting ((0,0)) gives \(0\leq 8\), which is true. Use a simple test point to decide the region quickly in exams.
Step 2
Why this answer is correct
The correct answer is B. ((0,0)). Substituting ((0,0)) gives \(0\leq 8\), which is true. Use a simple test point to decide the region quickly in exams.
Step 3
Exam Tip
((0,0)) रखने पर \(0\leq 8\) सत्य है। परीक्षा में सरल बिंदु से क्षेत्र जल्दी तय करें।
A. सीमा रेखा (x+y=5) है और टूटी हुई है/The boundary line is (x+y=5) and dashed
Step 1
Concept
The boundary line is made by replacing the inequality with equality, so it is (x+y=5), and it is dashed because of (<). In exams, first convert the inequality into an equation.
Step 2
Why this answer is correct
The correct answer is A. सीमा रेखा (x+y=5) है और टूटी हुई है / The boundary line is (x+y=5) and dashed. The boundary line is made by replacing the inequality with equality, so it is (x+y=5), and it is dashed because of (<). In exams, first convert the inequality into an equation.
Step 3
Exam Tip
सीमा रेखा बराबरी लगाकर (x+y=5) बनती है और (<) के कारण टूटी होती है। परीक्षा में पहले असमानता को समीकरण में बदलें।
C. ((0,0)) हल क्षेत्र में नहीं है/((0,0)) is not in the solution region
Step 1
Concept
Substituting ((0,0)) gives \(0\geq 6\), which is false, so the origin side is not the solution. In exams, shade the opposite side when the test is false.
Step 2
Why this answer is correct
The correct answer is C. ((0,0)) हल क्षेत्र में नहीं है / ((0,0)) is not in the solution region. Substituting ((0,0)) gives \(0\geq 6\), which is false, so the origin side is not the solution. In exams, shade the opposite side when the test is false.
Step 3
Exam Tip
((0,0)) रखने पर \(0\geq 6\) असत्य है, इसलिए मूल बिंदु वाला भाग हल नहीं है। परीक्षा में असत्य मिलने पर विपरीत भाग छायांकित करें।
D. रेखा (x=-2) के दाईं ओर, रेखा छोड़कर/Right of the line (x=-2), excluding the line
Step 1
Concept
In (x> -2), all (x)-values are greater than (-2), so shading is to the right. For inequalities in (x), check left or right in exams.
Step 2
Why this answer is correct
The correct answer is D. रेखा (x=-2) के दाईं ओर, रेखा छोड़कर / Right of the line (x=-2), excluding the line. In (x> -2), all (x)-values are greater than (-2), so shading is to the right. For inequalities in (x), check left or right in exams.
Step 3
Exam Tip
(x> -2) में सभी (x) मान (-2) से बड़े हैं, इसलिए दाईं ओर छायांकन होगा। परीक्षा में (x) वाली असमानता के लिए बायां-दायां क्षेत्र देखें।
B. रेखा (y=3) के नीचे और रेखा सहित/Below the line (y=3) including the line
Step 1
Concept
In \(y\leq 3\), the values of (y) are (3) or less. In exams, identify above or below for horizontal lines.
Step 2
Why this answer is correct
The correct answer is B. रेखा (y=3) के नीचे और रेखा सहित / Below the line (y=3) including the line. In \(y\leq 3\), the values of (y) are (3) or less. In exams, identify above or below for horizontal lines.
Step 3
Exam Tip
\(y\leq 3\) में (y) के मान (3) या उससे कम हैं। परीक्षा में क्षैतिज रेखा के लिए ऊपर-नीचे क्षेत्र पहचानें।
Since (>) does not include equality, the boundary line is dashed. Remember dashed lines for (<) and (>) in exams.
Step 2
Why this answer is correct
The correct answer is B. टूटी हुई रेखा / Dashed line. Since (>) does not include equality, the boundary line is dashed. Remember dashed lines for (<) and (>) in exams.
Step 3
Exam Tip
क्योंकि (>) में बराबरी शामिल नहीं होती, इसलिए सीमा रेखा टूटी हुई होती है। परीक्षा में (<) और (>) के लिए टूटी रेखा याद रखें।
Substituting ((0,0)) gives \(0\leq 4\), so the point lies in the solution region. In exams, test a convenient point first.
Step 2
Why this answer is correct
The correct answer is C. हल क्षेत्र में / In solution region. Substituting ((0,0)) gives \(0\leq 4\), so the point lies in the solution region. In exams, test a convenient point first.
Step 3
Exam Tip
((0,0)) रखने पर \(0\leq 4\) सत्य है, इसलिए बिंदु हल क्षेत्र में है। परीक्षा में पहले परीक्षण बिंदु रखें।
Clearing denominators gives \(4x-22\le 1\), so \(x\le \frac{23}{4}\). In fractional inequalities, multiply by the LCM first.
Step 2
Why this answer is correct
The correct answer is B. \(x\le \frac{23}{4}\). Clearing denominators gives \(4x-22\le 1\), so \(x\le \frac{23}{4}\). In fractional inequalities, multiply by the LCM first.
Step 3
Exam Tip
हर हटाने पर \(4x-22\le 1\) मिलता है, इसलिए \(x\le \frac{23}{4}\)। भिन्नों वाली असमानता में पहले लघुत्तम समापवर्त्य से गुणा करें।
