mathematics-class-11-unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtlfw

### <Success>Solution</Success>
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<p><strong>for (i)</strong> $3 x+2 y=24$</p>
<p>put $y=0 \Rightarrow x=8$</p>
<p>$x=0 \Rightarrow y=12$</p>
<p>$\therefore$ points are $(8,0)$ & $(0,12)$</p>
<p><strong>for (ii)</strong> $\quad 3 x+y=15$</p>
<p>put $y=0 \Rightarrow x=5$</p>
<p>$x=0 \Rightarrow y=15$</p>
<p><strong>for (iii)</strong> $x=4$ is a line $\parallel$ to $y-$ axis passing through $(4,0)$</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-5a.jpg"></p>
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### <Success>Solution</Success>
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<p>Now for (i) i.e. $3 x+2 y \geqslant 24$</p>
<p><strong>origin test:</strong> put $x=0$ & $y=0$</p>
<p>$ 3 \times 0+2 \times 0=0 \ngeq 24 $</p>
<p>$ \therefore(0,0) \notin \text{solution region}.$</p>
<p><strong>for (ii) i.e.</strong> $3 x+y \leqslant 15$</p>
<p>put $x=0 \quad$ &$ ~y=0$</p>
<p>$3 \times 0+0=0 \leqslant 15$ (True)</p>
<p>$\therefore(0,0) \in $ solution region.</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-6.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-6a.jpg"></p>
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### <Success>Solution</Success>
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<p><strong>for (i)</strong> $ 2 x+3 y=12$</p>
<p>put $y=0 \Rightarrow x=6$</p>
<p>$x=0 \Rightarrow y=4$</p>
<p>$\therefore$ points are
$(6,0)$ & $(0,4)$</p>
<p><strong>for(ii)</strong> $ x=2$ is a line || to $y-\text{axis}$ and passing through $(2,0)$</p>
<p><strong>for (iii)</strong> $y=1$ is a line || to $ x-\text{axis}$ and passing through $(0,1)$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-9.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-9a.jpg"></p>
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### <Success>Solution</Success>
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<p>$\underline{\text{Origin test:}}$</p>
<p><strong>for (i)</strong> $2 x+3 y \leqslant 12$</p>
<p>put $x=0$ & $y=0$</p>
<p>$2 \times 0+3 \times 0=0 \leqslant 12$ (True)</p>
<p>$\therefore(0,0)$ lies in the solution region for $2 x+3 y \leqslant 12$</p>
<p><strong>for (ii)</strong> $x\geqslant 2 \ \ \Rightarrow \ $ solution region lies in right hand side of line $x=2$.</p>
<p><strong>for (iii)</strong> $y \geqslant 1 \ \ \Rightarrow \ $ solution region lies in upper part of the line $y = 1 $</p>
<p>Shaded region $ABC$ will satisfy all the three given constraints. Hence, solution region will be the shaded region $ABC.$</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-10a.jpg">
<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-10b.jpg"></p>
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### <Success>Solution</Success>
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<p>$\underline{\text{Origin test:}}$</p>
<p>for (i) $x+2 y \leqslant 8$</p>
<p>put $x=0, y=0 \Rightarrow 0+2 \times 0=0 \leqslant 8$ (True)</p>
<p>$\therefore(0,0) \in$ Solution Region.</p>
<p>for (ii) $2 x+y \leqslant 8$</p>
<p>put $x=0, y=0 \Rightarrow 2 \times 0+0=0 \leqslant 8$ (True)</p>
<p>$(0,0) \in$ Solution Region</p>
<p>Common solution Region for the given system of inequalities (i), (ii) & (iii) will be shaded region $O A B C.$</p>
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<img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-14a.jpg"></p>
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<p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/50973ae5-2eb3-4541-e1e2-3f2871b3b500/public"></p>

### <Success>Solution</Success>
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<p><strong>for (i)</strong>
$ x+y=5$</p>
<p>put $y=0 \Rightarrow x=5$</p>
<p>$x=0 \Rightarrow y=5$</p>
<p>$\therefore$ points are $(5,0)$  & $(0,5)$</p>
<p><strong>for (ii)</strong>
$ 4 x+y=4 $</p>
<p>$ \text { put } y=0 \Rightarrow x=1 $</p>
<h4 id="-x0-rightarrow-y4-">$ x=0 \Rightarrow y=4 $</h4>
<p>points are $(1,0)$ & $(0,4)$</p>
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<p>======</p>
<h3 id="successsolutionsuccess"><Success>Solution</Success></h3>
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<p><strong>for (iii)</strong></p>
<h4 id="x5-y5-">$x+5 y=5 $</h4>
<h4 id="text--put--y0-rightarrow-x5-">$\text { put } y=0 \Rightarrow x=5 $</h4>
<h4 id="x0-rightarrow-y1-">$x=0 \Rightarrow y=1 $</h4>
<p>points are $ (5,0)$ & $(0,1)$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/mathematics-class-11-Unit-07-chapter-04-linear-inequalities-l-4-6-joqrwadtLfw-17.jpg"></p>