Inequalities Practice Questions with Solution: 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. This article contains Inequalities Practice Questions with Solution , which are designed to sharpen your problem-solving skills.
What are Inequalities?
In mathematics, an inequalities are 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:
Inequality Name | Symbol | Expression | Description |
|---|---|---|---|
Greater than | > | x > a | x is greater than a |
Less than | < | x < a | x is less than a |
Greater than equal to | ≥ | x ≥ a | x is greater than or equal to a |
Less than equal to | ≤ | x ≤ a | x is less than or equal to a |
Not equal | ≠ | x ≠ a | x is not equal to a |
Inequalities Practice Questions with Solution
Here are some Inequalities Practice Questions with Solution, complete with examples, questions and answers. Let's dive into mastering the techniques to solve them effectively.
1. Linear Inequality: Solve the inequality: 3x+5>11
3x+5>11
Subtract 5 from both sides: 3x>6
Divide both sides by 3: x>2
So, the solution is x>2
2. Compound Inequality: Solve the compound inequality: -2<2x+3≤7
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
3. Absolute Value Inequality: Solve the inequality: ∣x-3∣≥4
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
4. Quadratic Inequality: Solve the inequality: x2-4x<3
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
5. Rational Inequality: Solve the inequality: (x-2) / (x+1) > 0
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
6. Exponential Inequality: Solve the inequality: 2x - 1 < 8
Rewrite 8 as a power of 2:
2x - 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 2x - 1 < 8 is x < 4
7. Logarithmic Inequality: Solve the inequality: log(x+1)>log(4)
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
8. Polynomial Inequality: Solve the polynomial inequality: x3 - 2x2 - 3x > 0
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
9. Solve the compound inequality: 1 < 2x + 3 < 7
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
10. Solve an inequality with absolute values: Solve ∣3x+2∣<7
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