mathematics-class-11-unit-07-chapter-01-linear-inequalities-l-1-6-ycst06admtu

##### <Success>Procedure to solve a linear inequality in one variable</Success>
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<p>(i) Simplify and collect like terms.</p>
<p>(ii) Remove fraction (or decimals) by multiplying both sides by an appropriate number/ factor . (i.e L.C.M of denominator or a power of 10 in case of decimals)</p>
<p>(iii) Isolate all variable terms an one side and all constants on other side.</p>
<p>(iv) Make the co-efficient of variable 1.</p>
<p>(v) Choose the solution set from the given replacement set.</p>
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### <Success>Example 1</Success>
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<p>Given $x \in \{-3,-4,-5,-6\}$ and $49 < 1 - 2x$ . Find the possible values of $x$. Also represent its solution set on the number line.</p>
<p><strong>Solution:</strong></p>
<p>Given inequation</p>
<p>$9 < 1 - 2x$</p>
<p>$\Rightarrow - 1 + 9 < -1 +1 - 2x$</p>
<p>$\Rightarrow 8 < - 2x$</p>
<p>$\Rightarrow - 8 > -(-2x)$</p>
<p>$\Rightarrow - 8 > 2x $</p>
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<h3 id="successsolutionsuccess"><Success>Solution</Success></h3>
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<p>$\Rightarrow 2x < - 8$
$\Rightarrow \frac {2x}{2} < \frac {-8}{2}$</p>
<p>$\Rightarrow x < -4$</p>
<p>Since replacement set is given as $x \in \{-3,-4,-5,-6\}$</p>
<p>& $x < -4$</p>
<p>solution set : $\{-5, -6\}$</p>
<p>For $9 < 1-2x$</p>
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### <Success>Example 2</Success>
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<p>Solve the inequality $3-2x \geq x-32$, given that</p>
<p>(i) $x \in \N$    (ii) $x \in W$    (iii) $x \in \Z$</p>
<p><strong>Solution</strong></p>
<p>Given inequation, $3-2x \geq x-32$</p>
<p>$\Rightarrow - 3 + 3 - 2x \geq - 3 + x - 32$</p>
<p>$\Rightarrow -2x \geq x - 35 \ \Rightarrow - 3x \geq - 35$</p>
<p>$\Rightarrow - ( - 3x) \leq -1 (-35)$</p>
<p>multiplying both side by -1</p>
<p>$\Rightarrow 3x \leq 35 \ \Rightarrow x \leq \frac {35}{3}$</p>
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### <Success>Example 5</Success>
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<p>Solve $\frac {2x + 1} {3} \geq \frac {3x-2}{5} , x\in \R$.</p>
<p>Graph the solution set on the number line.</p>
<p><strong>Solution</strong></p>
<p>Given inequation, $\frac {2x+1}{3} \geq \frac {3x-2}{5}$ (LCM of 3 & 5 = 15)</p>
<p>Multiply both side by 15</p>
<p>$\Rightarrow 15 (\frac {2x+1}{3}) \geq 15 (\frac {3x-2}{5})$</p>
<p>$\Rightarrow 5(2x+1) \geq 3(3x-2)$</p>
<p>$\Rightarrow 10x + 5 \geq 9x - 6$</p>
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### <Success>Example 7</Success>
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<p>A man wants to cut three length from a single piece of board of length 91cm , the second length is to be 3cm longer than the shortest and the third length is to be twice as long as the shortest . What are the possible length of the shortest board if the third piece is to be at least 5cm longer than the second.</p>
<p><strong>Solution</strong></p>
<p>Let the length of shortest piece $= x$ cm</p>
<p>$\therefore$ Length of second piece $= x + 3$</p>
<p>Length of third piece $= 2x$</p>
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### <Success>Solution</Success>
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<p>$x + (x+3) + 2x \leq 91$</p>
<p>$\Rightarrow 4x + 3 \leq 91$</p>
<p>$\Rightarrow 4x + 3 - 3 \leq 91-3$</p>
<p>$\Rightarrow 4x \leq 88$</p>
<p>$\Rightarrow \frac {4x}{4} \leq \frac{88}{4}$</p>
<p>$\Rightarrow x \leq 22 \longrightarrow$ (i)</p>
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<p>======</p>
<h3 id="successsolutionsuccess-1"><Success>Solution</Success></h3>
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<p>again, $2x \geq (x+3) + 5$</p>
<p>$\Rightarrow 2x \geq x + 8$</p>
<p>$\Rightarrow 2x - x \geq x + 8 - x$</p>
<p>$\Rightarrow x \geq 8 \longrightarrow$ (ii)</p>
<p>from (i) & (ii)</p>
<p>$8 \leq x \leq 22$</p>
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### <Success>Solution</Success>
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<p>$\Rightarrow 2x + 2 -2 < 23 - 2$</p>
<p>$\Rightarrow 2x < 21$</p>
<p>$\Rightarrow x < \frac {21}{2} \longrightarrow$ (ii)</p>
<p>We have $x>5$ & $x < \frac{21}{2}$</p>
<p>$\Rightarrow 5 < x < \frac{21}{2}$</p>
<p>$\Rightarrow 5 < x < 10.5$</p>
<p>$\therefore$ We can take even number 6,8,10</p>
<p>$\therefore$ possible pairs are (6,8), (8,10) & (10,12).</p>
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<p>======</p>
<h3 id="successthank-yousuccess"><Success>Thank you</Success></h3>
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