Solving Two-Step Inequalities

In this lesson, we will learn how to solve two-step inequalities. Two-step inequalities are equations that require two different operations to isolate the variable. This is similar to solving two-step equations, but we need to account for the direction of the inequality sign.

Outline

• Introduction to Two-Step Inequalities
• Examples and Solutions
• Practice Exercises
• Homework
• Revision
• YouTube Video

Examples and Solutions

Example 1

Solve the inequality: \( 2x + 3 < 7 \)

Subtract 3 from both sides: \( 2x < 4 \)

Divide both sides by 2: \( x < 2 \)

Solution: \( x < 2 \)

Example 2

Solve the inequality: \( \frac{x}{4} - 5 \geq 3 \)

Add 5 to both sides: \( \frac{x}{4} \geq 8 \)

Multiply both sides by 4: \( x \geq 32 \)

Solution: \( x \geq 32 \)

Example 3

Solve the inequality: \( -3x + 6 < 0 \)

Subtract 6 from both sides: \( -3x < -6 \)

Divide both sides by -3 (and reverse the inequality sign): \( x > 2 \)

Solution: \( x > 2 \)

Example 4

Solve the inequality: \( 5 - \frac{x}{2} \leq 1 \)

Subtract 5 from both sides: \( -\frac{x}{2} \leq -4 \)

Multiply both sides by -2 (and reverse the inequality sign): \( x \geq 8 \)

Solution: \( x \geq 8 \)

Example 5

Solve the inequality: \( 7x - 4 > 10 \)

Add 4 to both sides: \( 7x > 14 \)

Divide both sides by 7: \( x > 2 \)

Solution: \( x > 2 \)

Example 6

Solve the inequality: \( -2x + 3 \leq -1 \)

Subtract 3 from both sides: \( -2x \leq -4 \)

Divide both sides by -2 (and reverse the inequality sign): \( x \geq 2 \)

Solution: \( x \geq 2 \)

Practice Exercises

Solve the following inequalities:

1. Solve \( 3x - 7 > 8 \)
2. Solve \( \frac{2x}{5} + 1 \leq 4 \)
3. Solve \( -4x + 9 > 5 \)
4. Solve \( 6 - \frac{x}{3} \geq 2 \)
5. Solve \( 5x - 4 < 11 \)
6. Solve \( -\frac{x}{2} + 7 \leq 3 \)

Solutions to Exercises

Solutions:

1. Solve \( 3x - 7 > 8 \)
   Add 7: \( 3x > 15 \)
   Divide by 3: \( x > 5 \)
2. Solve \( \frac{2x}{5} + 1 \leq 4 \)
   Subtract 1: \( \frac{2x}{5} \leq 3 \)
   Multiply by 5: \( 2x \leq 15 \)
   Divide by 2: \( x \leq 7.5 \)
3. Solve \( -4x + 9 > 5 \)
   Subtract 9: \( -4x > -4 \)
   Divide by -4 (reverse sign): \( x < 1 \)
4. Solve \( 6 - \frac{x}{3} \geq 2 \)
   Subtract 6: \( -\frac{x}{3} \geq -4 \)
   Multiply by -3 (reverse sign): \( x \leq 12 \)
5. Solve \( 5x - 4 < 11 \)
   Add 4: \( 5x < 15 \)
   Divide by 5: \( x < 3 \)
6. Solve \( -\frac{x}{2} + 7 \leq 3 \)
   Subtract 7: \( -\frac{x}{2} \leq -4 \)
   Multiply by -2 (reverse sign): \( x \geq 8 \)

Homework

Complete the following problems for additional practice:

1. Solve \( 2x + 5 < 11 \)
2. Solve \( \frac{3x - 6}{2} \geq 4 \)
3. Solve \( -5x + 7 \leq -3 \)
4. Solve \( \frac{x + 3}{4} > 2 \)
5. Solve \( 4 - \frac{2x}{5} < 1 \)
6. Solve \( -3x + 8 \geq 2 \)

Revision

In this lesson, we learned how to solve two-step inequalities by isolating the variable through two operations. Make sure to remember to reverse the inequality sign when multiplying or dividing by a negative number.

YouTube Video

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