GMAT Inequalities Questions

Expression 1 : $$5x - 2$$ > $$3x - 4$$

=> $$5x - 3x$$ > $$2 - 4$$

=> $$2x$$ > $$-2$$

=> $$x$$ > $$-1$$ ----------(i)

Expression 2 : $$3x + 4(1 - x)$$ > $$5x - 2$$

=> $$-x + 4$$ > $$5x - 2$$

=> $$5x + x$$ < $$4 + 2$$

=> $$6x$$ < $$6$$

=> $$x$$ < $$1$$ ------(ii)

Combining inequalities (i) and (ii), we get : $$-1$$ < $$x$$ < $$1$$

Thus, only value that $$x$$ can take = 0

=> Ans - (C)

Expression 1 : 4 + 3x ≤ 6 + x

=> $$3x - x \leq 6 - 4$$ => $$2x \leq 2$$

=> $$x \leq \frac{2}{2}$$ => $$x \leq 1$$ --------------(i)

Expression 2 : 3x + 5 > 2 + 2x

=> $$3x - 2x $$ > $$ 2 - 5$$

=> $$x $$ > $$ -3$$ ------------(ii)

Combining inequalities (i) and (ii), we get : $$-3 $$ < $$ x \leq 1$$

Thus, $$x$$ can take values = -2,-1,0,1

=> Ans - (B)

Expression 1 : $$2 + 2x < 3 + 5x$$

=> $$5x - 2x$$ > $$2 - 3$$

=> $$x$$ > $$\frac{-1}{3}$$ ----------(i)

Expression 2 : $$3(x - 2)2 < 5 - x$$

=> $$6x - 12$$ < $$5 - x$$

=> $$6x + x$$ < $$5 + 12$$

=> $$x$$ < $$\frac{17}{7}$$ ------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{-1}{3}$$ < $$x$$ < $$\frac{17}{7}$$

Thus, $$x$$ can take values = 0 , 1 , 2

=> Ans - (A)

Expression 1 : 7 + 4x > 3 + 3x

=> $$4x - 3x$$ > $$3 - 7$$

=> $$x$$ > $$-4$$ --------------(i)

Expression 2 : 3x - 2 < 5 - x

=> $$3x + x$$ < $$5 + 2$$

=> $$4x$$ < $$7$$

=> $$x$$ < $$\frac{7}{4}$$ -----------------(ii)

Combining inequalities (i) and (ii), we get : $$-4$$ < $$x$$ < $$\frac{7}{4}$$

Thus, $$x$$ can take values = -3 , -2 , -1 , 0 , 1

=> Ans - (C)

Expression 1 : $$3 - x < 2x + 5$$

=> $$2x + x$$ > $$3 - 5$$

=> $$x$$ > $$\frac{-2}{3}$$ ----------(i)

Expression 2 : $$4(x + 1) - 3 < 3 - x$$

=> $$4x + 1$$ < $$3 - x$$

=> $$4x + x$$ < $$3 - 1$$

=> $$x$$ < $$\frac{2}{5}$$ ------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{-2}{3}$$ < $$x$$ < $$\frac{2}{5}$$

Thus, only value that $$x$$ can take among the options = 0

=> Ans - (B)

Expression 1 : 5 ­- 5x < 4 ­- x

=> $$5x-x$$ > $$5-4$$

=> $$4x$$ > $$1$$

=> $$x$$ > $$\frac{1}{4}$$ ------------(i)

Expression 2 : 2 ­- x < 6 -­ 4x

=> $$4x-x$$ < $$6-2$$

=> $$3x$$ < $$4$$

=> $$x$$ < $$\frac{4}{3}$$ --------------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{1}{4}$$ < $$x$$ < $$\frac{4}{3}$$

The only value that $$x$$ can take among the given options = 1

=> Ans - (A)

Given, $$5\left(4-x\right)-4<3x-2>4x-6$$

$$=$$>  $$20-5x-4<3x-2>4x-6$$

$$=$$>  $$16-5x<3x-2>4x-6$$

$$=$$>  $$16-5x<3x-2$$  and  $$3x-2>4x-6$$

$$=$$>  $$8x>18$$  and  $$x<4$$

$$=$$>  $$x>\frac{\ 9}{4}$$  and  $$x<4$$

$$=$$>  $$x>\ 2.25$$  and  $$x<4$$

$$=$$>  $$2.25<x<4$$

$$\therefore\ x=3$$

Expression 1 : 5x + 5 > 2 + 2x

=> $$5x - 2x$$ > $$2 - 5$$

=> $$3x$$ > $$-3$$

=> $$x$$ > $$-1$$ -------------(i)

Expression 2 : 5x + 3 ≤ 4x + 5

=> $$5x - 4x \leq 5 - 3$$

=> $$x \leq 2$$ ------------(ii)

Combining inequalities (i) and (ii), => $$-1$$ < $$x \leq 2$$

Thus, the values that $$x$$ can take = 0 , 1 , 2

=> Ans - (D)

Expression 1 : 5x - 4 ≤ 2 - x

=> $$5x + x \leq 2 + 4$$ => $$6x \leq 6$$

=> $$x \leq 1$$ ---------(i)

Expression 2 : 4x + 5 > 2x - 5

=> $$4x - 2x$$ > $$-5 -5$$ => $$2x$$ > $$-10$$

=> $$x$$ > $$-5$$ ----------(ii)

Combining inequalities (i) and (ii), we get : $$-5$$ < $$x \leq 1$$

Thus, $$x$$ can take values = -4 , -3 , -2 , -1 , 0 , 1

=> Ans - (D)

Expression 1 : 2x - 1 < 5x + 2

=> $$5x - 2x$$ > $$-1 - 2$$ => $$3x$$ > $$-3$$

=> $$x$$ > $$\frac{-3}{3}$$ => $$x$$ > $$-1$$ --------------(i)

Expression 2 : 2x + 5 < 6 - 3x

=> $$2x + 3x$$ < $$6 - 5$$ => $$5x$$ < $$1$$

=> $$x$$ < $$\frac{1}{5}$$ ------------(ii)

Combining equations (i) and (ii), we get : $$-1$$ < $$x$$ < $$\frac{1}{5}$$

Thus, the only possible value that $$x$$ can take = 0

=> Ans - (B)

Expression 1 : 2x -­ 3 ≤ 5 + x

=> $$2x-x \leq 5+3$$

=> $$x \leq 8$$ ------------(i)

Expression 2 : 5 -­ x < 1 + 5x

=> $$5x+x$$ > $$5-1$$

=> $$6x$$ > $$4$$

=> $$x$$ > $$\frac{2}{3}$$ -----------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{2}{3}$$ < $$x \leq 8$$

The only value that $$x$$ can take among the options = 1

=> Ans - (B)

Expression 1 : 5 ­- 3x < 4 ­- x

=> $$3x-x$$ > $$5-4$$

=> $$2x$$ > $$1$$

=> $$x$$ > $$\frac{1}{2}$$ ------------(i)

Expression 2 : 5(2 -­ x) > 2 -­ 2x

=> $$10-5x$$ > $$2-2x$$

=> $$5x-2x$$ < $$10-2$$

=> $$3x$$ < $$8$$

=> $$x$$ < $$\frac{8}{3}$$ --------------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{1}{2}$$ < $$x$$ < $$\frac{8}{3}$$

The only value that $$x$$ can take among the given options = 1

=> Ans - (C)

Expression 1 : 2(3x + 5) > 4x -­ 5

=> $$6x+10$$ > $$4x-5$$

=> $$6x-4x$$ > $$-5-10$$

=> $$2x$$ > $$-15$$

=> $$x$$ > $$\frac{-15}{2}$$ --------(i)

Expression 2 : 4x -­ 5 < 3x + 2

=> $$4x-3x$$ < $$2+5$$

=> $$x$$ < $$7$$ ---------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{-15}{2}$$ < $$x$$ < $$7$$

The only value that $$x$$ can take = 6

=> Ans - (B)

Expression 1 : 3x + 2 < 2x + 1

=> $$3x-2x$$ < $$1-2$$

=> $$x$$ < $$-1$$ ------------(i)

Expression 2 : x -­ 4 ≤ 2x -­ 1

=> $$2x-x \geq -4+1$$

=> $$x \geq -3$$ ------------(ii)

Thus, combining inequalities (i) and (ii), we get : $$-3 \leq x$$ < $$-1$$

The only possible value that $$x$$ can take among the given options = -2

=> Ans - (A)

Expression 1 : 4(4x + 5) > 2x - 1

=> $$16x+20$$ > $$2x-1$$

=> $$16x-2x$$ > $$-1-20$$

=> $$14x$$ > $$-21$$

=> $$x$$ > $$\frac{-3}{2}$$ -----------(i)

Expression 2 : 2x - 1 > 4x - 3

=> $$4x-2x$$ < $$3-1$$

=> $$2x$$ < $$2$$

=> $$x$$ < $$1$$ ----------(ii)

Combining inequalities (i) and (ii), we get : $$\frac{-3}{2}$$ < $$x$$ < $$1$$

The only value that $$x$$ can take among the options = 0

=> Ans - (D)