Section 3.1: Solving Multi-Step Inequalities

Multi-step inequalities require more than one operation to isolate the variable. The rules are similar to equations, with one key exception:

  • When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol.

Example 1

Solve \( 3x + 5 < 11 \)

Step 1: Subtract 5 from both sides: \( 3x < 6 \)

Step 2: Divide by 3: \( x < 2 \)

Solution: \( x < 2 \)

Example 2 (Negative Coefficient)

Solve \( -2x + 4 \ge 10 \)

Step 1: Subtract 4 from both sides: \( -2x \ge 6 \)

Step 2: Divide by -2 and reverse inequality: \( x \le -3 \)

Solution: \( x \le -3 \)

Practice Problems

  1. Solve: \( 5x - 7 < 18 \)
  2. Solve: \( 4x + 9 \ge 1 \)
  3. Solve: \( -3x + 2 < 8 \)
  4. Solve: \( 2 - 5x > 12 \)
  5. Solve: \( -4x - 6 \le 10 \)
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