Solving Inequalities Practice Questions

Inequalities are fundamental in mathematics and are used extensively in various branches such as algebra, calculus, optimization and more. They describe relationships between variables and constants and are essential in solving problems involving ranges of values, constraints and conditions.

Solving inequalities often involves manipulating expressions algebraically, graphing on number lines or coordinate planes and understanding intervals of possible solutions. The solutions to inequalities are sets of values or intervals that satisfy the given inequality condition.

What is Inequality?

In mathematics, an inequality is a statement that compares two expressions or values, indicating their relative sizes. Unlike equations, which assert equality between two expressions, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

Example: 2x + 5 < 1 = This inequality states that 2 times some unknown number x, plus 5, is strictly less than 1.

Unlike equations where both sides aim to be equal, inequalities establish a relationship between expressions using symbols like:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

Formally, an inequality is typically written in one of the following forms:

Solving Inequalities Practice Questions

Question 1: Linear Inequality: Solve the inequality: 3x+5>11

Solution:

3x+5>11

Subtract 5 from both sides: 3x>6

Divide both sides by 3: x>2

So, the solution is x>2

Question 2: Compound Inequality: Solve the compound inequality: -2<2x+3≤7

Solution:

Break it into two parts:

-2<2x+3 and 2x+3 ≤ 7

-2<2x+3

Subtract 3 from all parts: -5<2x

Divide by 2:

-5/2 <x

2x+3 ≤ 7

Subtract 3 from all parts:

2x ≤ 4

Divide by 2:

x ≤2

So, the solution to -2 < 2x+3 ≤ 7 is -5/2 <x ≤2

Question 3: Absolute Value Inequality: Solve the inequality: ∣x-3∣≥4

Solution:

Consider two cases:

(i) x-3≥4

Add 3 to both sides: x ≥ 7

(ii) x-3≤-4

Add 3 to both sides:

x ≤ -1

So, the solution to ∣x-3∣ ≥ 4 is x≤-1 or x ≥7

Question 4: Quadratic Inequality: Solve the inequality: x²-4x<3

Solution:

Rewrite it as: x²-4x-3<0

Factorize the quadratic expression: (x-3)(x+1)<0

Analyze the sign changes: The inequality holds when -1<x<3

So, the solution to x² – 4x <3 is -1 < x < 3

Question 5: Rational Inequality: Solve the inequality: (x-2) / (x+1) > 0

Solution:

Consider where the numerator and denominator change signs:

• Numerator (x-2) changes sign at x = 2
• Denominator (x+1) changes sign at x = -1

Test intervals:

• For x<-1 or -1 <x<2, (x-2) / (x+1) <0
• For x > 2 or -1 < x < 2, (x-2) / (x+1) >0

So, the solution for (x-2)/(x+1) > 0 is x <-1 or x> 2

Question 6: Exponential Inequality: Solve the inequality: 2^x – 1 < 8

Solution:

Rewrite 8 as a power of 2:

2^x – 1 < 2³

Since the bases are the same, equate the exponents:

x-1 < 3

Add 1 to both sides:

x < 4

So, the solution to 2^x – 1 < 8 is x < 4

Question 7: Logarithmic Inequality: Solve the inequality: log⁡(x+1)>log⁡(4)

Solution:

Since the logarithm function is increasing:

log⁡(x+1) > log⁡(4)

x + 1 > 4

Subtract 1 from both sides: x > 3

So, the solution to log⁡(x+1) > log⁡(4) is x > 3

Question 8: Polynomial Inequality: Solve the polynomial inequality: x³ – 2x² – 3x > 0

Solution:

Factorize the polynomial (if possible) or analyze intervals:

x³ – 2x² – 3x > 0

x(x-3)(x+1) > 0

Determine sign changes and intervals:

Inequality holds when x < -1 or x > 3

So , the solution for x³ – 2x² – 3x > 0 is x <-1 or x > 3

Question 9: Solve the compound inequality: 1 < 2x + 3 < 7

Solution:

Split this into two separate inequalities:

1 < 2x+3 and 2x+3 < 7

Solve the first inequality:

1 < 2x + 3

Subtract 3 from both sides:

-2 < 2x

Divide both sides by 2:

-1 < x

Solve the second inequality

2x+3 < 7

Subtract 3 from both sides:

2x < 4

Divide both sides by 2:

x < 2

So, the solution set for the compound inequality 1 < 2x+3 < 7 is -1 < x < 2

Question 10: Solve an inequality with absolute values: Solve ∣3x+2∣<7

Solution:

Consider two cases for the absolute value inequality:

Case 1: 3x+2 < 7

3x < 5

x < 5/3

Case 2: -(3x+2) < 7

-3x-2 < 7

Subtract 2 from both sides:

-3x < 9

Divide both sides by -3 (flip the inequality sign):

x > -3

Combining the solutions from both cases, we get -3 < x < 5/3

Also Read:

• Triangle Inequality theorem
• Linear Inequalities
• Compound Inequalities

Solving Inequalities – FAQs

What is Concept of Inequality in Maths?

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

What is the Most Important thing to Remember when Solving Inequalities?

If you multiply or divide both sides of an inequality by the same positive number, the inequality remains true. But if you multiply or divide both sides of an inequality by a negative number, the inequality is no longer true. In fact, the inequality becomes reversed.

What is Difference Between an Equation and an Inequality?

Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

What is Theorem of Inequality?

According to triangle inequality theorem, for any given triangle, the sum of two sides of a triangle is always greater than the third side. A polygon bounded by three line-segments is known as the Triangle.

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