Inequalities are mathematical statements comparing two expressions, showing that one is greater than, greater than or equal to, less than, or less than or equal to the other. Understanding how to solve inequalities is crucial for various mathematical applications and problem-solving scenarios. This guide will walk you through the process, covering different types and offering practical examples.
Understanding Inequality Symbols
Before diving into solving techniques, let's refresh our memory on the inequality symbols:
- > Greater than
- ≥ Greater than or equal to
- < Less than
- ≤ Less than or equal to
- ≠ Not equal to
Solving Linear Inequalities
Linear inequalities involve variables raised to the power of 1. The process of solving them is similar to solving linear equations, with one crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.
Example 1: Solve 2x + 5 > 9
- Subtract 5 from both sides: 2x > 4
- Divide both sides by 2: x > 2
The solution is x > 2, meaning any value of x greater than 2 satisfies the inequality.
Example 2: Solve -3x + 6 ≤ 12
- Subtract 6 from both sides: -3x ≤ 6
- Divide both sides by -3 and reverse the inequality sign: x ≥ -2
The solution is x ≥ -2. Notice how the inequality sign flipped because we divided by a negative number.
Solving Compound Inequalities
Compound inequalities involve two or more inequalities combined using "and" or "or."
Example 3 (And): Solve -2 ≤ 3x - 5 < 7
- Add 5 to all parts of the inequality: 3 ≤ 3x < 12
- Divide all parts by 3: 1 ≤ x < 4
The solution is 1 ≤ x < 4, meaning x can be any value between 1 and 4, inclusive of 1 but not 4.
Example 4 (Or): Solve x + 2 > 5 or 2x - 1 < -3
- Solve the first inequality: x + 2 > 5 => x > 3
- Solve the second inequality: 2x - 1 < -3 => 2x < -2 => x < -1
The solution is x > 3 or x < -1.
Solving Polynomial Inequalities
Polynomial inequalities involve variables raised to higher powers. The steps for solving them are:
- Rewrite the inequality with zero on one side.
- Find the roots (solutions) of the corresponding equation.
- Test values in the intervals created by the roots to determine the solution.
Example 5: Solve x² - 4x + 3 > 0
- Factor the quadratic: (x - 1)(x - 3) > 0
- Find the roots: x = 1 and x = 3
- Test intervals:
- x < 1: (negative)(negative) > 0 (True)
- 1 < x < 3: (positive)(negative) > 0 (False)
- x > 3: (positive)(positive) > 0 (True)
The solution is x < 1 or x > 3.
Solving Absolute Value Inequalities
Absolute value inequalities involve the absolute value function | |. Remember that |x| represents the distance of x from 0.
Example 6: Solve |x - 2| < 3
This inequality means the distance between x and 2 is less than 3. We can rewrite this as a compound inequality:
-3 < x - 2 < 3
Solving this gives: -1 < x < 5
Example 7: Solve |x + 1| ≥ 4
This means the distance between x and -1 is greater than or equal to 4. This translates to two separate inequalities:
x + 1 ≥ 4 or x + 1 ≤ -4
Solving these gives: x ≥ 3 or x ≤ -5
Graphing Inequalities
Graphing inequalities helps visualize the solution set. For linear inequalities, you'll typically use a number line, shading the region representing the solution. For inequalities with two variables, you'll use a coordinate plane. Remember to use an open circle (o) for > and < and a closed circle (•) for ≥ and ≤.
By mastering these techniques, you'll be well-equipped to solve a wide range of inequalities, significantly enhancing your mathematical problem-solving skills. Remember to practice regularly to build confidence and fluency.