\(0+0\le 6\) is true, so the side containing origin is shaded. In exams, always use a test point.
Step 2
Why this answer is correct
The correct answer is A. मूल बिंदु वाली ओर / Side containing origin. \(0+0\le 6\) is true, so the side containing origin is shaded. In exams, always use a test point.
Step 3
Exam Tip
\(0+0\le 6\) सत्य है, इसलिए मूल बिंदु वाली ओर छायांकित होगी। परीक्षा में हमेशा परीक्षण बिंदु लगाएं।
Since the inequality has (>), the boundary line is not included. Remember dashed lines for (<) and (>).
Step 2
Why this answer is correct
The correct answer is A. टूटी हुई रेखा / Dashed line. Since the inequality has (>), the boundary line is not included. Remember dashed lines for (<) and (>).
Step 3
Exam Tip
क्योंकि असमिका में (>) है, सीमा रेखा शामिल नहीं होती। परीक्षा में (<) और (>) के लिए टूटी रेखा याद रखें।
A. पहले चतुर्थांश में सीमित त्रिभुज/Bounded triangle in first quadrant
Step 1
Concept
The axes and the line (x+y=8) form a bounded triangle. Identifying vertices is useful in exams.
Step 2
Why this answer is correct
The correct answer is A. पहले चतुर्थांश में सीमित त्रिभुज / Bounded triangle in first quadrant. The axes and the line (x+y=8) form a bounded triangle. Identifying vertices is useful in exams.
Step 3
Exam Tip
दोनों अक्षों और रेखा (x+y=8) से सीमित त्रिभुज बनता है। परीक्षा में vertices पहचानना उपयोगी है।
Putting (y=0) gives (x=3), and putting (x=0) gives (y=4). The intercept method is quick for graphing.
Step 2
Why this answer is correct
The correct answer is A. ((3,0)) और ((0,4)) / ((3,0)) and ((0,4)). Putting (y=0) gives (x=3), and putting (x=0) gives (y=4). The intercept method is quick for graphing.
Step 3
Exam Tip
(y=0) रखने पर (x=3) और (x=0) रखने पर (y=4) मिलता है। परीक्षा में intercept method graph के लिए तेज है।
A. क्योंकि \(0-0\ge 2\) असत्य है/Because \(0-0\ge 2\) is false
Step 1
Concept
Substituting origin makes the inequality false. Use a test point to choose the correct half-plane.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(0-0\ge 2\) असत्य है / Because \(0-0\ge 2\) is false. Substituting origin makes the inequality false. Use a test point to choose the correct half-plane.
Step 3
Exam Tip
मूल बिंदु रखने पर असमिका गलत हो जाती है। परीक्षा में test point से सही half-plane चुनें।
A. पहला चतुर्थांश सहित अक्ष/First quadrant including axes
Step 1
Concept
Both (x) and (y) are non-negative, so the region is the first quadrant. Do not miss non-negative restrictions in exams.
Step 2
Why this answer is correct
The correct answer is A. पहला चतुर्थांश सहित अक्ष / First quadrant including axes. Both (x) and (y) are non-negative, so the region is the first quadrant. Do not miss non-negative restrictions in exams.
Step 3
Exam Tip
(x) और (y) दोनों non-negative हैं, इसलिए क्षेत्र पहला चतुर्थांश है। परीक्षा में non-negative restrictions को न भूलें।
It is impossible for (x+y) to be at most (2) and at least (5) at the same time. Identify contradictory conditions in exams.
Step 2
Why this answer is correct
The correct answer is A. रिक्त क्षेत्र / Empty region. It is impossible for (x+y) to be at most (2) and at least (5) at the same time. Identify contradictory conditions in exams.
Step 3
Exam Tip
एक ही समय में (x+y) का (2) से कम और (5) से अधिक होना संभव नहीं है। परीक्षा में विरोधी conditions पहचानें।
In the first quadrant, the region above (x+y=7) extends infinitely. Distinguish bounded and unbounded regions in exams.
Step 2
Why this answer is correct
The correct answer is A. असीमित क्षेत्र / Unbounded region. In the first quadrant, the region above (x+y=7) extends infinitely. Distinguish bounded and unbounded regions in exams.
Step 3
Exam Tip
पहले चतुर्थांश में रेखा (x+y=7) के ऊपर का क्षेत्र अनंत तक जाता है। परीक्षा में bounded और unbounded में अंतर करें।
A. हाँ, क्योंकि (2+2(3)\le 8)/Yes, because (2+2(3)\le 8)
Step 1
Concept
(2+6=8), so the point lies on the boundary and is included. Equality is allowed in \(\le\).
Step 2
Why this answer is correct
The correct answer is A. हाँ, क्योंकि (2+2(3)\le 8) / Yes, because (2+2(3)\le 8). (2+6=8), so the point lies on the boundary and is included. Equality is allowed in \(\le\).
Step 3
Exam Tip
(2+6=8), इसलिए बिंदु boundary पर है और शामिल है। परीक्षा में equality आने पर भी \(\le\) में बिंदु मान्य होता है।
Both (x) and (y) lie in bounded intervals, so a rectangle is formed. Use vertical and horizontal lines for such inequalities.
Step 2
Why this answer is correct
The correct answer is A. आयत / Rectangle. Both (x) and (y) lie in bounded intervals, so a rectangle is formed. Use vertical and horizontal lines for such inequalities.
Step 3
Exam Tip
(x) और (y) दोनों सीमित intervals में हैं, इसलिए आयत बनता है। परीक्षा में ऐसी असमिकाओं को vertical और horizontal lines से बनाएं।
In (x<4), (x) is less than (4), so the left side is chosen. For vertical lines, think left-right.
Step 2
Why this answer is correct
The correct answer is A. रेखा के बाईं ओर / Left of the line. In (x<4), (x) is less than (4), so the left side is chosen. For vertical lines, think left-right.
Step 3
Exam Tip
(x<4) में (x) का मान (4) से छोटा होता है, इसलिए बायां क्षेत्र चुना जाता है। परीक्षा में vertical line के लिए left-right सोचें।
For \(y\ge -2\), the boundary is included and the region above it is taken. For horizontal lines, check above-below.
Step 2
Why this answer is correct
The correct answer is A. रेखा के ऊपर सहित / On and above the line. For \(y\ge -2\), the boundary is included and the region above it is taken. For horizontal lines, check above-below.
Step 3
Exam Tip
\(y\ge -2\) में boundary शामिल है और ऊपर का भाग लिया जाता है। परीक्षा में horizontal line के लिए above-below देखें।
A. मूल बिंदु वाली ओर सही है/The side containing origin is correct
Step 1
Concept
\(-0+0\le 1\) is true, so the side containing origin is shaded. Choose a simple test point in exams.
Step 2
Why this answer is correct
The correct answer is A. मूल बिंदु वाली ओर सही है / The side containing origin is correct. \(-0+0\le 1\) is true, so the side containing origin is shaded. Choose a simple test point in exams.
Step 3
Exam Tip
\(-0+0\le 1\) सत्य है, इसलिए मूल बिंदु वाली ओर shading होगी। परीक्षा में सरल test point चुनें।
A. कोई ऐसा बिंदु जो रेखा पर न हो/Any point not lying on the line
Step 1
Concept
A point on the boundary does not identify the shaded side. Take an easy point away from the line.
Step 2
Why this answer is correct
The correct answer is A. कोई ऐसा बिंदु जो रेखा पर न हो / Any point not lying on the line. A point on the boundary does not identify the shaded side. Take an easy point away from the line.
Step 3
Exam Tip
boundary पर स्थित बिंदु shading side नहीं बताता। परीक्षा में रेखा से बाहर का आसान बिंदु लें।
For any (x), (y) cannot be both below (x+1) and above (x+4). Check parallel constraints in exams.
Step 2
Why this answer is correct
The correct answer is A. रिक्त क्षेत्र / Empty region. For any (x), (y) cannot be both below (x+1) and above (x+4). Check parallel constraints in exams.
Step 3
Exam Tip
किसी (x) के लिए (y) एक साथ (x+1) से कम और (x+4) से अधिक नहीं हो सकता। परीक्षा में parallel constraints check करें।
A. दो समानांतर रेखाओं के बीच की पट्टी/Strip between two parallel lines
Step 1
Concept
Both lines have the same slope and the region lies between them. Use equal slopes to identify parallel lines.
Step 2
Why this answer is correct
The correct answer is A. दो समानांतर रेखाओं के बीच की पट्टी / Strip between two parallel lines. Both lines have the same slope and the region lies between them. Use equal slopes to identify parallel lines.
Step 3
Exam Tip
दोनों रेखाओं की slope समान है और क्षेत्र इनके बीच आता है। परीक्षा में समान slope से parallel lines पहचानें।
A. boundary पर और हल में/On boundary and in solution
Step 1
Concept
(2(5)+5(0)=10), so the point is on the boundary and included for \(\ge\). Check equality carefully.
Step 2
Why this answer is correct
The correct answer is A. boundary पर और हल में / On boundary and in solution. (2(5)+5(0)=10), so the point is on the boundary and included for \(\ge\). Check equality carefully.
Step 3
Exam Tip
(2(5)+5(0)=10), इसलिए बिंदु boundary पर है और \(\ge\) में शामिल है। परीक्षा में equality को ध्यान से देखें।
Because of \(y\ge\), the region above the line is taken and the line is included. The direction is clear in (y)-form.
Step 2
Why this answer is correct
The correct answer is A. रेखा के ऊपर / Above the line. Because of \(y\ge\), the region above the line is taken and the line is included. The direction is clear in (y)-form.
Step 3
Exam Tip
\(y\ge\) के कारण रेखा के ऊपर का क्षेत्र लिया जाता है और रेखा शामिल होती है। परीक्षा में (y)-form में दिशा तुरंत दिखती है।
A. (x=5) पूर्ण और (y=1) टूटी/(x=5) solid and (y=1) dashed
Step 1
Concept
\(\le\) gives a solid line and (>) gives a dashed line. Check each boundary type separately.
Step 2
Why this answer is correct
The correct answer is A. (x=5) पूर्ण और (y=1) टूटी / (x=5) solid and (y=1) dashed. \(\le\) gives a solid line and (>) gives a dashed line. Check each boundary type separately.
Step 3
Exam Tip
\(\le\) में solid line और (>) में dashed line होती है। परीक्षा में हर inequality का boundary type अलग जांचें।
The small bounded region with axes is in the first quadrant and below the line. Use the origin test in exams.
Step 2
Why this answer is correct
The correct answer is A. \(x\ge 0\), \(y\ge 0\), \(x+2y\le 4\). The small bounded region with axes is in the first quadrant and below the line. Use the origin test in exams.
Step 3
Exam Tip
axes के साथ छोटा bounded region पहले चतुर्थांश में और रेखा के नीचे होता है। परीक्षा में origin test करें।
\(3x-y\le 0\) gives \(y\ge 3x\). Converting to (y)-form makes graphing easier.
Step 2
Why this answer is correct
The correct answer is A. \(y\ge 3x\) का क्षेत्र / Region of \(y\ge 3x\). \(3x-y\le 0\) gives \(y\ge 3x\). Converting to (y)-form makes graphing easier.
Step 3
Exam Tip
\(3x-y\le 0\) से \(y\ge 3x\) मिलता है। परीक्षा में inequality को (y)-form में बदलना आसान होता है।
The region is a \(2\times 3\) rectangle, so its area is (6). Recognize bounded rectangles quickly.
Step 2
Why this answer is correct
The correct answer is A. (6) वर्ग इकाई / (6) square units. The region is a \(2\times 3\) rectangle, so its area is (6). Recognize bounded rectangles quickly.
Step 3
Exam Tip
क्षेत्र \(2\times 3\) का आयत है, इसलिए क्षेत्रफल (6) है। परीक्षा में bounded rectangle को तुरंत पहचानें।
The intercepts are (4) and (4), so area is \(\frac{1}{2}\times 4\times 4=8\). Remember the intercept triangle formula.
Step 2
Why this answer is correct
The correct answer is A. (8) वर्ग इकाई / (8) square units. The intercepts are (4) and (4), so area is \(\frac{1}{2}\times 4\times 4=8\). Remember the intercept triangle formula.
Step 3
Exam Tip
त्रिभुज के intercepts (4) और (4) हैं, इसलिए क्षेत्रफल \(\frac{1}{2}\times 4\times 4=8\) है। परीक्षा में intercept triangle का formula याद रखें।
A. मूल बिंदु वाली ओर/On the side containing origin
Step 1
Concept
The half-plane containing the true test point is the solution side. This is the quickest graphical check.
Step 2
Why this answer is correct
The correct answer is A. मूल बिंदु वाली ओर / On the side containing origin. The half-plane containing the true test point is the solution side. This is the quickest graphical check.
Step 3
Exam Tip
सत्य test point जिस half-plane में होता है वही solution side होती है। परीक्षा में यह सबसे तेज graphical check है।
At origin the value is (0), so the origin side is \(\le 6\) and the opposite side is \(\ge 6\). Use equality signs when boundary is included.
Step 2
Why this answer is correct
The correct answer is A. \(2x+3y\ge 6\). At origin the value is (0), so the origin side is \(\le 6\) and the opposite side is \(\ge 6\). Use equality signs when boundary is included.
Step 3
Exam Tip
मूल बिंदु पर (0) मिलता है, इसलिए origin वाली ओर \(\le 6\) है और विपरीत ओर \(\ge 6\) होगी। परीक्षा में boundary included हो तो equality वाला चिन्ह लें।
The boundary is (x-2y=6), giving (x=6) and (y=-3). Draw the equality line before the inequality.
Step 2
Why this answer is correct
The correct answer is A. ((6,0)) और ((0,-3)) / ((6,0)) and ((0,-3)). The boundary is (x-2y=6), giving (x=6) and (y=-3). Draw the equality line before the inequality.
Step 3
Exam Tip
boundary (x-2y=6) है, इससे (x=6) और (y=-3) मिलता है। परीक्षा में inequality से पहले equality line बनाएं।
((2,1)) is on the boundary because (2(2)+1=5), but a strict inequality excludes the boundary. Do not include equal points for (<).
Step 2
Why this answer is correct
The correct answer is A. ((2,1)). ((2,1)) is on the boundary because (2(2)+1=5), but a strict inequality excludes the boundary. Do not include equal points for (<).
Step 3
Exam Tip
((2,1)) boundary पर है क्योंकि (2(2)+1=5), लेकिन strict असमिका boundary को शामिल नहीं करती। परीक्षा में (<) में equal point न लें।
Pairwise intersections of (x=2), (y=1), and (x+y=6) give these vertices. Solve boundary lines pairwise in exams.
Step 2
Why this answer is correct
The correct answer is A. ((2,1)), ((5,1)), ((2,4)). Pairwise intersections of (x=2), (y=1), and (x+y=6) give these vertices. Solve boundary lines pairwise in exams.
Step 3
Exam Tip
(x=2), (y=1), और (x+y=6) के प्रतिच्छेदों से ये तीन vertices मिलते हैं। परीक्षा में boundary lines pairwise solve करें।
Below the line gives \(y\le\), and including the boundary includes equality. Remember \(y\le\) for below the line.
Step 2
Why this answer is correct
The correct answer is A. \(y\le 9-3x\). Below the line gives \(y\le\), and including the boundary includes equality. Remember \(y\le\) for below the line.
Step 3
Exam Tip
नीचे का क्षेत्र \(y\le\) देता है और boundary शामिल होने से equality भी शामिल है। परीक्षा में नीचे के लिए \(y\le\) याद रखें।
The region lies inside the square \(0\le x\le 4\), \(0\le y\le 4\) and above (x+y=4). Draw the larger rectangle first in exams.
Step 2
Why this answer is correct
The correct answer is A. सीमित बहुभुज / Bounded polygon. The region lies inside the square \(0\le x\le 4\), \(0\le y\le 4\) and above (x+y=4). Draw the larger rectangle first in exams.
Step 3
Exam Tip
क्षेत्र वर्ग \(0\le x\le 4\), \(0\le y\le 4\) के भीतर और (x+y=4) के ऊपर है। परीक्षा में पहले बड़ा rectangle बनाकर कटे भाग पहचानें।
Substituting ((6,0)) gives \(12\le 10\), which is false, so the point is outside. Use substitution for point location.
Step 2
Why this answer is correct
The correct answer is A. हल क्षेत्र के बाहर / Outside the solution region. Substituting ((6,0)) gives \(12\le 10\), which is false, so the point is outside. Use substitution for point location.
Step 3
Exam Tip
((6,0)) रखने पर \(12\le 10\) असत्य है, इसलिए बिंदु बाहर है। परीक्षा में point location के लिए substitution करें।
A. पहले सभी boundary lines बनाकर common shaded region लें/Draw all boundary lines first and take the common shaded region
Step 1
Concept
The graph of a system is the common shaded region. In exams, intersect the half-planes of all inequalities.
Step 2
Why this answer is correct
The correct answer is A. पहले सभी boundary lines बनाकर common shaded region लें / Draw all boundary lines first and take the common shaded region. The graph of a system is the common shaded region. In exams, intersect the half-planes of all inequalities.
Step 3
Exam Tip
system का graph common shaded region से मिलता है। परीक्षा में हर inequality का half-plane काटकर intersection लें।
