Lesson 3.4 Solving Two-Step and multi-step Inequalities

Solve inequalities that contain more than one operation

Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time.

Step 1: Undo addition or subtraction

Step 2: Undo multiplication and division

Is your variable isolated?

Subtracting a number is the same as adding its opposite.7 – 2t = 7 + (–2t)

Remember!

Solving Multi-Step InequalitiesSolve the inequality and graph the solutions.

45 + 2b > 6145 + 2b > 61

–45 –452b > 16

b > 8

0 2 4 6 8 10 12 14 16 18 20

1. Add/ Subtract

2. Multiply/Divide

The solution set is {b:b > 8}.

8 – 3y ≥ 298 – 3y ≥ 29

–8 –8

–3y ≥ 21

y ≤ –7

1. Add/Subtract

2. Multiply/Divide

–10 –8 –6 –4 –2 0 2 4 6 8 10

–7

Solving Multi-Step InequalitiesSolve the inequality and graph the solutions.

The solution set is {y:y –7}.

Solve and Graph the inequality

1) 9 2b 13

2)2x 13 9

3) y5

128

4) 3m 5 17

To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides.

Simplifying Before Solving Inequalities

Solve the inequality and graph the solutions.2 – (–10) > –4t

12 > –4t

–3 < t (or t > –3)

1. Combine like terms.

2. Multiply/Divide

–3

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {t:t > –3}.

Simplifying Before Solving InequalitiesSolve 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

1. Distributive Property

2. Add/Subtract

3. Multiply/Divide

–10 –8 –6 –4 –2 0 2 4 6 8 10

The solution set is {x:x ≤ 4}.

Now you try…

x > 3

x ≥ -1

x ≤ 2

x < 35

x ≥ -27

1. 3x – 7 > 2 4. x – 4 < 3 5

2. 4x + 1 -3 5. 15 + x ≥ 6

33. 2x – 7 ≤ -3

Lesson Quiz: Part ISolve each inequality and graph the solutions.

1. 13 – 2x ≥ 21 x ≤ –4

2. –11 + 2 < 3p p > –3

3. 23 < –2(3 – t) t > 7

4.

Lesson Quiz: Part II

5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A? more than 12 movies