Solving Multi-Step Inequalities
In this lesson, we will explore how to solve multi-step inequalities. These inequalities involve more than one step to isolate the variable. We'll cover techniques to solve these inequalities and check solutions to ensure they are correct.
Outline
- Introduction to Multi-Step Inequalities
- Solving Inequalities with Addition and Subtraction
- Solving Inequalities with Multiplication and Division
- Examples and Practice Exercises
- Homework
- Revision
- YouTube Video Explanation
Examples
Example 1
Solve the inequality: \(2x - 5 > 7\)
Step 1: Add 5 to both sides: \(2x > 12\)
Step 2: Divide by 2: \(x > 6\)
The solution is \(x > 6\).
Example 2
Solve the inequality: \(\frac{3x}{4} \leq 9\)
Step 1: Multiply both sides by 4: \(3x \leq 36\)
Step 2: Divide by 3: \(x \leq 12\)
The solution is \(x \leq 12\).
Example 3
Solve the inequality: \(-4x + 7 < 3\)
Step 1: Subtract 7 from both sides: \(-4x < -4\)
Step 2: Divide by -4 (remember to reverse the inequality sign): \(x > 1\)
The solution is \(x > 1\).
Example 4
Solve the inequality: \(5 - 2x \geq 9\)
Step 1: Subtract 5 from both sides: \(-2x \geq 4\)
Step 2: Divide by -2 (reverse the inequality sign): \(x \leq -2\)
The solution is \(x \leq -2\).
Example 5
Solve the inequality: \( \frac{2x - 1}{3} > 4\)
Step 1: Multiply both sides by 3: \(2x - 1 > 12\)
Step 2: Add 1 to both sides: \(2x > 13\)
Step 3: Divide by 2: \(x > 6.5\)
The solution is \(x > 6.5\).
Example 6
Solve the inequality: \( -3x - 4 \leq 8\)
Step 1: Add 4 to both sides: \(-3x \leq 12\)
Step 2: Divide by -3 (reverse the inequality sign): \(x \geq -4\)
The solution is \(x \geq -4\).
Exercises
Exercise 1
Solve the inequality: \(3x + 2 > 11\)
Step 1: Subtract 2 from both sides: \(3x > 9\)
Step 2: Divide by 3: \(x > 3\)
The solution is \(x > 3\).
Exercise 2
Solve the inequality: \(-\frac{4x}{5} \leq -8\)
Step 1: Multiply both sides by -5/4 (reverse the inequality sign): \(x \geq 10\)
The solution is \(x \geq 10\).
Exercise 3
Solve the inequality: \(2x - 7 < 5\)
Step 1: Add 7 to both sides: \(2x < 12\)
Step 2: Divide by 2: \(x < 6\)
The solution is \(x < 6\).
Exercise 4
Solve the inequality: \( -6 + 2x \geq -2\)
Step 1: Add 6 to both sides: \(2x \geq 4\)
Step 2: Divide by 2: \(x \geq 2\)
The solution is \(x \geq 2\).
Exercise 5
Solve the inequality: \(\frac{5x + 1}{2} < 3\)
Step 1: Multiply both sides by 2: \(5x + 1 < 6\)
Step 2: Subtract 1 from both sides: \(5x < 5\)
Step 3: Divide by 5: \(x < 1\)
The solution is \(x < 1\).
Exercise 6
Solve the inequality: \(4 - 3x > -2\)
Step 1: Subtract 4 from both sides: \(-3x > -6\)
Step 2: Divide by -3 (reverse the inequality sign): \(x < 2\)
The solution is \(x < 2\).
Homework
Complete the following exercises for additional practice:
- Solve the inequality: \(7x - 3 \leq 11\)
- Solve the inequality: \(\frac{-3x + 4}{2} > 5\)
- Solve the inequality: \(-2x + 6 \geq 4\)
- Solve the inequality: \(\frac{5 - x}{3} < 2\)
Revision
Review the following key points:
- When solving inequalities, perform the same operations on both sides to isolate the variable.
- Remember to reverse the inequality sign when multiplying or dividing by a negative number.
- Check your solution by substituting it back into the original inequality to ensure it satisfies the condition.
Video Explanation
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