3.4 Solving Two-Step and Multi-Step Inequalities

Algebra 4.0, 5.0

Solve inequalities that contain more than one operation.

Main Idea

• • • When we solve multi-step equations: – We use more than one operation – We use inverse operations – We may need to combine like terms – We may need to use the distributive property – We may need to multiply reciprocals to get rid of fractions All these items hold true for inequalities What do we need to be careful of?

Two-Step Inequalities: Practice 1) -12 > 3x + 6

3 )

x

 2  5  3

2) 8 – 3y > 29

x

4 ) 3  4   4

Example-Solving Multi-Step Inequalities • Solve and graph solution

Example: Distributive Property Solve the inequality and graph the solutions.

–4(2 – x) ≤ 8

−4 (2 – x) ≤ 8 −4 (2) − 4 (−x) ≤ 8 –8 + 4x ≤ 8 +8 +8 4x ≤ 16 x ≤ 4

Distribute Since –4 on the left side.

–8 is added to 4x, add 8 to both sides.

Since x is multiplied by 4, divide both sides by 4 to undo the multiplication.

The solution set is { x:x ≤ 4 } .

–10 –8 –6 –4 –2 0 2 4 6 8 10

Example: Distributive Property & Combine Like Terms Solve the inequality and graph the solutions. Check your answer.

3 + 2(x + 4) > 3

Distribute 2 on the left side.

3 + 2 (x + 4) > 3 3 + 2x + 8 > 3 2x + 11 > 3 – 11 – 11 2x > –8 x > –4

Combine like terms.

Since 11 is added to 2x, subtract 11 from both sides to undo the addition.

Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.

The solution set is { x:x > –4 } .

–10 –8 –6 –4 –2 0 2 4 6 8 10

Multi-Step Practice

• Solve and graph solution.

Example-Simplify before Solving

• Solve and graph solutions

Example-Simplify before Solving

• Solve and graph solutions

Example-Simplify before Solving

• Solve and graph solutions

Practice

• Solve and graph solutions

Review

1) What is important to remember when solving inequalities?

2) What is difficult when solving multi-step inequalities?