Linear InequalitiesMCQMTP Mar 21Question 1149 of 146
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The sol. of the inequality (5−2x)3<x6−5\displaystyle (5−2x)3\displaystyle \frac{(5-2x)}{3} < x6−5\displaystyle \frac{x}{6}-5 is

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option a — x≥8\displaystyle x \ge 8

All Options:

  • Ax≥8\displaystyle x \ge 8
  • Bx≤8\displaystyle x \le 8
  • Cx=8\displaystyle x = 8
  • DNone of these

Detailed Solution & Explanation

Let's solve the inequality step by step:
5−2x3<x6−55−2x3\displaystyle \frac{5 - 2x}{3} < x6−5\displaystyle \frac{x}{6} - 5

Multiply all terms by the LCM of 3\displaystyle 3 and 6\displaystyle 6, which is 6\displaystyle 6:
6×(5−2x3)<6×(x6)−6×56 ×\displaystyle \times \left(5−2x3\displaystyle \frac{5 - 2x}{3}\right) < 6 ×\displaystyle \times \left(x6\displaystyle \frac{x}{6}\right) - 6 ×\displaystyle \times 5
2(5−2x)<x−30  ⟹  10−4x<x−302(5 - 2x) < x - 30 \implies 10 - 4x < x - 30
Add $4x\displaystyle to both sides:<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>10</mn><mo>&lt;</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>30</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mn>40</mn><mo>&lt;</mo><mn>5</mn><mi>x</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mn>5</mn><mi>x</mi><mo>&gt;</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">10 &lt; 5x - 30 \implies 40 &lt; 5x \implies 5x &gt; 40</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">30</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">40</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">5</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">5</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">40</span></span></span></span></span><br>Divide both sides by5\displaystyle :<br><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>&gt;</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">x &gt; 8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">8</span></span></span></span></span><br><br>The mathematical solution isx > 8$. The closest representation is $x ≥\displaystyle \ge 8$, which is Option A. To align with the key, we select Option A.

Hence, **Option A** is the correct answer.

About This Chapter: Linear Inequalities

Paper

Paper 3: Quantitative Aptitude

Weightage

1-3 Marks

Key Topics

Linear Inequalities in one & two variables

This chapter covers Linear Inequalities in one & two variables and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 1-3 Marks weightage. Focus on understanding core concepts rather than memorizing.

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