JEE Mathematics Linear Inequalities MCQs Set 04

Download JEE MCQs for JEE Mathematics: Linear Inequalities

Access targeted multiple-choice questions for Linear Inequalities designed to align with the latest JEE academic syllabus for JEE Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

Chapter-wise Objective Questions: Linear Inequalities

Access the complete set of multiple-choice questions for Linear Inequalities below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.

Multiple Choice Questions

Question 1: If x is a real number and \( |x|<3 \), then
(a) \( x\ge3 \)
(b) \( -3<x<3 \)
(c) \( x\le-3 \)
(d) \( -3\le x\le3 \)
Show Answer & Explanation

Answer: (b) \( -3<x<3 \)

Explanation:
1. \( |x|<a \) with \( a>0 \) means \( -a<x<a \).
2. So \( -3<x<3 \).

Teacher's Note:
1. Strict inequality gives an open interval.

Question 2: Given that x, y and b are real numbers and \( x<y \), \( b<0 \), then
(a) \( \dfrac xb<\dfrac yb \)
(b) \( \dfrac xb\le\dfrac yb \)
(c) \( \dfrac xb>\dfrac yb \)
(d) \( \dfrac xb\ge\dfrac yb \)
Show Answer & Explanation

Answer: (c) \( \dfrac xb>\dfrac yb \)

Explanation:
1. Dividing both sides of an inequality by a negative number reverses its direction.
2. So \( x<y \) gives \( \frac xb>\frac yb \).

Teacher's Note:
1. Reverse the sign when dividing by a negative.

Question 3: If \( |x+3|\ge10 \), then
(a) \( x\in(-13,7] \)
(b) \( x\in(-13,7) \)
(c) \( x\in(-\infty,13]\cup[-7,\infty) \)
(d) \( x\in(-\infty,-13]\cup[7,\infty) \)
Show Answer & Explanation

Answer: (d) \( x\in(-\infty,-13]\cup[7,\infty) \)

Explanation:
1. \( |x+3|\ge10 \) means \( x+3\ge10 \) or \( x+3\le-10 \).
2. So \( x\ge7 \) or \( x\le-13 \).

Teacher's Note:
1. A modulus greater than a number splits into two rays.

Question 4: Let \( \dfrac C5=\dfrac{F-32}{9} \). If C lies between 10 and 20, then
(a) \( 50<F<78 \)
(b) \( 50<F<68 \)
(c) \( 49<F<68 \)
(d) \( 49<F<78 \)
Show Answer & Explanation

Answer: (b) \( 50<F<68 \)

Explanation:
1. \( F=\frac95C+32 \).
2. For \( C=10 \), \( F=50 \); for \( C=20 \), \( F=68 \). So \( 50<F<68 \).

Teacher's Note:
1. F is an increasing function of C.

Question 5: The solution set of the inequality \( 4x+3<6x+7 \) is
(a) \( [-2,\infty) \)
(b) \( (-\infty,-2) \)
(c) \( (-2,\infty) \)
(d) none of these
Show Answer & Explanation

Answer: (c) \( (-2,\infty) \)

Explanation:
1. \( 3-7<6x-4x \), so \( -4<2x \).
2. \( x>-2 \).

Teacher's Note:
1. Collect x terms on one side.

Question 6: Which of the following is the solution set of \( 3x-7>5x-1 \) for \( x\in\mathbb R \)?
(a) \( (-\infty,-3) \)
(b) \( (-\infty,-3] \)
(c) \( (-3,\infty) \)
(d) \( (-3,3) \)
Show Answer & Explanation

Answer: (a) \( (-\infty,-3) \)

Explanation:
1. \( -7+1>5x-3x \) gives \( -6>2x \).
2. \( x<-3 \).

Teacher's Note:
1. Dividing by 2 keeps the direction.

Question 7: The solution set of the inequality \( 37-(3x+5)\ge9x-8(x-3) \) is
(a) \( (-\infty,2) \)
(b) \( (-\infty,-2) \)
(c) \( (-\infty,-2] \)
(d) \( (-\infty,2] \)
Show Answer & Explanation

Answer: (d) \( (-\infty,2] \)

Explanation:
1. Left side: \( 32-3x \). Right side: \( 9x-8x+24=x+24 \).
2. \( 32-3x\ge x+24 \) gives \( 8\ge4x \), so \( x\le2 \).

Teacher's Note:
1. Expand each side first.

Question 8: The solution set of the inequalities \( 6\le-3(2x-4)<12 \) is
(a) \( (-\infty,1] \)
(b) \( (0,1] \)
(c) \( (0,1]\cup[1,\infty) \)
(d) \( [1,\infty) \)
Show Answer & Explanation

Answer: (b) \( (0,1] \)

Explanation:
1. \( 6\le-6x+12<12 \) gives \( -6\le-6x<0 \).
2. Dividing by \( -6 \) reverses the signs: \( 1\ge x>0 \), i.e. \( (0,1] \).

Teacher's Note:
1. Reverse the inequalities when dividing by -6.

Question 9: Which of the following is the solution set of the inequalities \( 2(x-1)<x+5 \) and \( 3(x+2)>2-x \)?
(a) \( (-\infty,-1) \)
(b) \( (-1,1) \)
(c) \( (-1,7) \)
(d) \( (1,7) \)
Show Answer & Explanation

Answer: (c) \( (-1,7) \)

Explanation:
1. First: \( 2x-2<x+5 \) gives \( x<7 \).
2. Second: \( 3x+6>2-x \) gives \( 4x>-4 \), \( x>-1 \). Together \( (-1,7) \).

Teacher's Note:
1. Intersect the two solution sets.

Question 10: x and b are real numbers. If \( b>0 \) and \( |x|>b \), then
(a) \( x\in(-b,\infty) \)
(b) \( x\in(-\infty,b) \)
(c) \( x\in(-b,b) \)
(d) \( x\in(-\infty,-b)\cup(b,\infty) \)
Show Answer & Explanation

Answer: (d) \( x\in(-\infty,-b)\cup(b,\infty) \)

Explanation:
1. \( |x|>b \) means \( x>b \) or \( x<-b \).
2. So \( x\in(-\infty,-b)\cup(b,\infty) \).

Teacher's Note:
1. The distance from 0 exceeds b.

Question 11: If \( a<b \) and \( c<0 \), then
(a) \( \dfrac ac=\dfrac bc \)
(b) \( \dfrac ac\le\dfrac bc \)
(c) \( \dfrac ac>\dfrac bc \)
(d) none of these
Show Answer & Explanation

Answer: (c) \( \dfrac ac>\dfrac bc \)

Explanation:
1. Dividing by a negative number reverses the inequality: \( \frac ac>\frac bc \).

Teacher's Note:
1. Reverse the sign for a negative divisor.

Question 12: The solution set of the system \( 3x-7<5+x \) and \( 11-5x\le1 \) is
(a) \( [2,6) \)
(b) \( (2,6] \)
(c) \( [2,6] \)
(d) \( (2,6) \)
Show Answer & Explanation

Answer: (a) \( [2,6) \)

Explanation:
1. \( 3x-7<5+x \) gives \( 2x<12 \), so \( x<6 \).
2. \( 11-5x\le1 \) gives \( -5x\le-10 \), so \( x\ge2 \). Together \( [2,6) \).

Teacher's Note:
1. Intersect \( x<6 \) with \( x\ge2 \).

Question 13: The solution set of the inequalities \( 3x-7>2(x-6) \) and \( 6-x>11-2x \) is
(a) \( (-5,\infty) \)
(b) \( [5,\infty) \)
(c) \( (5,\infty) \)
(d) \( [-5,\infty) \)
Show Answer & Explanation

Answer: (c) \( (5,\infty) \)

Explanation:
1. First: \( 3x-7>2x-12 \) gives \( x>-5 \).
2. Second: \( -x+2x>11-6 \) gives \( x>5 \). Both hold when \( x>5 \).

Teacher's Note:
1. Take the common part of the two rays.

Question 14: If \( \dfrac{5-2x}{3}\le\dfrac x6-5 \), then \( x\in \)
(a) \( [2,\infty) \)
(b) \( [-8,8] \)
(c) \( [4,\infty) \)
(d) \( [8,\infty) \)
Show Answer & Explanation

Answer: (d) \( [8,\infty) \)

Explanation:
1. Multiply by 6: \( 2(5-2x)\le x-30 \), i.e. \( 10-4x\le x-30 \).
2. \( 40\le5x \), so \( x\ge8 \).

Teacher's Note:
1. Clear the denominators by multiplying by 6.

Question 15: If \( \dfrac{3x-4}{2}\ge\dfrac{x+1}{4}-1 \), then \( x\in \)
(a) \( [1,\infty) \)
(b) \( (1,\infty) \)
(c) \( (-5,5) \)
(d) \( [-5,5] \)
Show Answer & Explanation

Answer: (a) \( [1,\infty) \)

Explanation:
1. Multiply by 4: \( 2(3x-4)\ge x+1-4 \), i.e. \( 6x-8\ge x-3 \).
2. \( 5x\ge5 \), so \( x\ge1 \).

Teacher's Note:
1. Multiply through by 4.

Question 16: If \( -5\le\dfrac{5-3x}{2}\le8 \), then \( x\in \)
(a) \( \left[-\dfrac{11}{3},5\right] \)
(b) \( [-5,5] \)
(c) \( \left[-\dfrac{11}{3},\infty\right) \)
(d) \( (-\infty,\infty) \)
Show Answer & Explanation

Answer: (a) \( \left[-\dfrac{11}{3},5\right] \)

Explanation:
1. Multiply by 2: \( -10\le5-3x\le16 \).
2. Subtract 5: \( -15\le-3x\le11 \); dividing by \( -3 \) gives \( -\frac{11}{3}\le x\le5 \).

Teacher's Note:
1. Reverse the signs when dividing by -3.

Question 17: The inequality \( \dfrac2x<3 \) is true when x belongs to
(a) \( \left[\dfrac23,\infty\right) \)
(b) \( \left(-\infty,\dfrac23\right] \)
(c) \( (-\infty,0)\cup\left(\dfrac23,\infty\right) \)
(d) none of these
Show Answer & Explanation

Answer: (c) \( (-\infty,0)\cup\left(\dfrac23,\infty\right) \)

Explanation:
1. For \( x>0 \): \( 2<3x \) gives \( x>\frac23 \).
2. For \( x<0 \): \( 2>3x \), which is always true. So \( x<0 \) or \( x>\frac23 \).

Teacher's Note:
1. Treat positive and negative x separately.

Question 18: The solution of \( |3x+2|<1 \) is
(a) \( \left[-1,-\dfrac13\right] \)
(b) \( \left\{-\dfrac13,-1\right\} \)
(c) \( \left(-1,-\dfrac13\right) \)
(d) none of these
Show Answer & Explanation

Answer: (c) \( \left(-1,-\dfrac13\right) \)

Explanation:
1. \( -1<3x+2<1 \) gives \( -3<3x<-1 \).
2. \( -1<x<-\frac13 \).

Teacher's Note:
1. Split the modulus into a double inequality.

Question 19: The solution of \( |x-1|\ge|x-3| \) is
(a) \( x\le2 \)
(b) \( x\ge2 \)
(c) \( [1,3] \)
(d) none of these
Show Answer & Explanation

Answer: (b) \( x\ge2 \)

Explanation:
1. Both sides are non-negative, so square: \( (x-1)^2\ge(x-3)^2 \).
2. \( -2x+1\ge-6x+9 \) gives \( 4x\ge8 \), so \( x\ge2 \).

Teacher's Note:
1. Square both sides.

Question 20: If \( -3x+17<-13 \), then
(a) \( x\in(10,\infty) \)
(b) \( x\in[10,\infty) \)
(c) \( x\in(-\infty,10] \)
(d) \( x\in[-10,10) \)
Show Answer & Explanation

Answer: (a) \( x\in(10,\infty) \)

Explanation:
1. \( -3x<-30 \).
2. Dividing by \( -3 \) reverses the sign: \( x>10 \).

Teacher's Note:
1. Reverse the sign when dividing by a negative.

Question 21: If \( |x+2|\le9 \), then
(a) \( x\in(-7,11) \)
(b) \( x\in[-11,7] \)
(c) \( x\in(-\infty,-7)\cup(11,\infty) \)
(d) \( x\in(-\infty,-7)\cup[11,\infty) \)
Show Answer & Explanation

Answer: (b) \( x\in[-11,7] \)

Explanation:
1. \( -9\le x+2\le9 \).
2. \( -11\le x\le7 \).

Teacher's Note:
1. A non-strict modulus bound gives a closed interval.

Question 22: Consider: I. Two real numbers or two algebraic expressions related by the symbols \( <,>,\le,\ge \) form an inequality. II. When equal numbers are added to (or subtracted from) both sides of an inequality, the inequality does not change. III. When both sides of an inequality are multiplied (or divided) by the same positive number, the inequality does not change. Which are true?
(a) Only I
(b) Only II
(c) Only III
(d) All of the above
Show Answer & Explanation

Answer: (d) All of the above

Explanation:
1. All three are the standard properties of inequalities.

Teacher's Note:
1. Recall the rules of inequalities.

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Multiple Choice Questions (MCQs) for JEE Mathematics Linear Inequalities

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