Inequalities in one Variable MCQ Quiz in தமிழ் - Objective Question with Answer for Inequalities in one Variable - இலவச PDF ஐப் பதிவிறக்கவும்
Last updated on Apr 7, 2025
Latest Inequalities in one Variable MCQ Objective Questions
Top Inequalities in one Variable MCQ Objective Questions
Inequalities in one Variable Question 1:
Find the solution of the inequality:
\(\frac{x}{4}< \frac{(5x-2)}{3}-\frac{(7x-3)}{5}\)
Answer (Detailed Solution Below)
Inequalities in one Variable Question 1 Detailed Solution
Concept:
Rules for Operations on Inequalities:
- Adding the same number to each side of an inequality does not change the direction of the inequality symbol.
- Subtracting the same number from each side of an inequality does not change the direction of the inequality symbol.
- Multiplying each side of an inequality by a positive number does not change the direction of the inequality symbol.
- Multiplying each side of an inequality by a negative number reverses the direction of the inequality symbol.
- Dividing each side of an inequality by a positive number does not change the direction of the inequality symbol.
- Dividing each side of an inequality by a negative number reverses the direction of the inequality symbol.
Calculation:
Given:
Let us rewrite the given inequality, we get
\(\frac{x}{4}< \frac{(5x-2)}{3}-\frac{(7x-3)}{5}\)
\(\Rightarrow \frac{x}{4}< \frac{5(5x-2)-3(7x-3)}{15}\)
\(\Rightarrow \frac{x}{4}< \frac{25x-10-21x+9}{15}\)
\(\Rightarrow \frac{x}{4}< \frac{4x-1}{15}\)
⇒ 15x < 4(4x - 1)
⇒ 15x < 16x - 4
⇒ 15x - 16x < 16x - 16x - 4
⇒ -x < -4
⇒ x > 4
Therefore, we can say that the solution set for the given inequality is \((4,\infty )\).
Inequalities in one Variable Question 2:
The solution set of the inequality 17 - (2x + 4) ≤ 9x - 4(2x - 3) is
Answer (Detailed Solution Below)
Inequalities in one Variable Question 2 Detailed Solution
Calculation:
We have, 17 - (2x + 4) ≤ 9x - 4(2x - 3)
⇒ 17 - 2x - 4 ≤ 9x - 8x + 12
⇒ 17 - 4 - 12 ≤ 9x - 8x + 2x
⇒ 1 ≤ 3x
⇒ x ≥ \(\frac{1}{3}\)
∴ The solution set is x ∈ \(\left[\frac{1}{3},\infty\right)\) .
Inequalities in one Variable Question 3:
The number of positive integral solutions of the inequation \(\frac{x + 2}{x + 3} > 1 \)
Answer (Detailed Solution Below)
Inequalities in one Variable Question 3 Detailed Solution
Given:
The inequation: (x + 2)/(x + 3) > 1
Formula used:
For a rational inequality of the form (a/b) > 1, we analyze critical points and test values between intervals.
Calculation:
(x + 2)/(x + 3) > 1
⇒ (x + 2) - (x + 3) > 0
⇒ x + 2 - x - 3 > 0
⇒ -1 > 0
This is not possible.
Since the inequality is never satisfied, there are no positive integral solutions.
∴ The correct answer is option (4).
Inequalities in one Variable Question 4:
The solution set of the inequality 37 − (3x + 5) ≥ 9x − 8(x − 3) is
Answer (Detailed Solution Below)
Inequalities in one Variable Question 4 Detailed Solution
Calculation
Given;
Inequality: 37 - (3x + 5) ≥ 9x - 8(x - 3)
⇒ 37 - 3x - 5 ≥ 9x - 8x + 24
⇒ 32 - 3x ≥ x + 24
⇒ 32 - 24 ≥ x + 3x
⇒ 8 ≥ 4x
⇒ 4x ≤ 8
⇒ x ≤ 2
∴ The solution set is (-∞, 2].
Hence option 3 is correct.
Inequalities in one Variable Question 5:
If x satisfies the inequality \(-3<\frac{1}{2}+\frac{-3 x}{2} \leq 6\), then x lies in the interval
Answer (Detailed Solution Below)
Inequalities in one Variable Question 5 Detailed Solution
Calculation
Given:
\(-3 < \frac{1}{2} + \frac{-3x}{2} \leq 6\)
⇒ \(-6 < 1 - 3x \leq 12\)
⇒ \(-7 < -3x \leq 11\)
⇒ \(\frac{-7}{-3} > x \geq \frac{11}{-3}\)
⇒ \(\frac{7}{3} > x \geq -\frac{11}{3}\)
⇒ \(-\frac{11}{3} \leq x < \frac{7}{3}\)
∴ x lies in the interval \(\left[-\frac{11}{3}, \frac{7}{3}\right)\)
Hence option 1 is correct
Inequalities in one Variable Question 6:
The solution set of the inequality 17 - (2x + 4) ≤ 9x - 4(2x - 3) is
Answer (Detailed Solution Below)
Inequalities in one Variable Question 6 Detailed Solution
Calculation:
We have, 17 - (2x + 4) ≤ 9x - 4(2x - 3)
⇒ 17 - 2x - 4 ≤ 9x - 8x + 12
⇒ 17 - 4 - 12 ≤ 9x - 8x + 2x
⇒ 1 ≤ 3x
⇒ x ≥ \(\frac{1}{3}\)
∴ The solution set is x ∈ \(\left[\frac{1}{3},\infty\right)\) .
Inequalities in one Variable Question 7:
If |3x - 5| ≤ 2 then
Answer (Detailed Solution Below)
Inequalities in one Variable Question 7 Detailed Solution
Concept:
If |x| ≤ a then - a ≤ x ≤ a
Calculations:
Given , |3x - 5| ≤ 2
⇒ - 2 ≤ 3x - 5 ≤ 2
⇒ - 2 + 5 ≤ 3x ≤ 2 + 5
⇒ 3 ≤ 3x ≤ 7
⇒\(\rm 1 \le x \le \dfrac{7}{3}\)
Hence, if |3x - 5| ≤ 2 then then \(\rm 1 \le x \le \dfrac{7}{3}\)
Inequalities in one Variable Question 8:
The solution set of the inequality 17 - (2x + 4) ≤ 9x - 4(2x - 3) is
Answer (Detailed Solution Below)
Inequalities in one Variable Question 8 Detailed Solution
Calculation:
We have, 17 - (2x + 4) ≤ 9x - 4(2x - 3)
⇒ 17 - 2x - 4 ≤ 9x - 8x + 12
⇒ 17 - 4 - 12 ≤ 9x - 8x + 2x
⇒ 1 ≤ 3x
⇒ x ≥ \(\frac{1}{3}\)
∴ The solution set is x ∈ \(\left[\frac{1}{3},\infty\right)\) .
Inequalities in one Variable Question 9:
On the number line, the solution of system of inequalities \(\left\{ \begin{matrix} 5+x > 3x - 7 \\\ 11 - 5x \le 1 \end{matrix} \right.\) is represented
Answer (Detailed Solution Below)
Inequalities in one Variable Question 9 Detailed Solution
Explanation:
Given system of linear inequality is
\(\left\{ \begin{matrix} 5 + x > 3x - 7 \\\ 11 - 5x \le 1 \end{matrix} \right.\)
When 5 + x > 3x - 7 ⇒ 2x < 12 ⇒ x < 6
When 11 - 5x ≤ 1 ⇒ 5x ≥ 10 ⇒ x ≥ 2
On the number line, the solution is represented as below.
Inequalities in one Variable Question 10:
If |3x - 5| ≤ 2 then
Answer (Detailed Solution Below)
Inequalities in one Variable Question 10 Detailed Solution
Concept:
If |x| ≤ a then - a ≤ x ≤ a
Calculations:
Given , |3x - 5| ≤ 2
⇒ - 2 ≤ 3x - 5 ≤ 2
⇒ - 2 + 5 ≤ 3x ≤ 2 + 5
⇒ 3 ≤ 3x ≤ 7
⇒\(\rm 1 \le x \le \dfrac{7}{3}\)
Hence, if |3x - 5| ≤ 2 then then \(\rm 1 \le x \le \dfrac{7}{3}\)