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3.4 Solving Two-Step and Multi-Step Inequalities

Algebra 4.0, 5.0Solve inequalities that contain more than one operation.

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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?

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Two-Step Inequalities: Practice

1) -12 > 3x + 6

2) 8 – 3y > 29

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352

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x

x

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Example-Solving Multi-Step Inequalities

• Solve and graph solution

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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 +84x ≤ 16

x ≤ 4

Distribute –4 on the left side.

Since –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.

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {x:x ≤ 4}.

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

3 + 2(x + 4) > 3

3 + 2(x + 4) > 33 + 2x + 8 > 3

2x + 11 > 3– 11 – 11

2x > –8

x > –4

Distribute 2 on the left side.

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.

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {x:x > –4}.

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Multi-Step Practice

• Solve and graph solution.

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Example-Simplify before Solving

• Solve and graph solutions

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Example-Simplify before Solving

• Solve and graph solutions

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Example-Simplify before Solving

• Solve and graph solutions

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Practice

• Solve and graph solutions

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Review

1) What is important to remember when solving inequalities?

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