1. Objective
Solve inequalities that contain variable terms on both sides.

2. Example 1A: Solving Inequalities with Variables on Both Sides
Solve the inequality and graph the solutions.
y ≤ 4y + 18
y ≤ 4y + 18
–y –y
0 ≤ 3y + 18
To collect the variable terms on one side, subtract y from both sides.
Since 18 is added to 3y, subtract 18 from both sides to undo the addition.
– – 18
–18 ≤ 3y
Since y is multiplied by 3, divide both sides by 3 to undo the multiplication.
–6 ≤ y (or y  –6)
–10 –8 –6 –4 –2 2 4 6 8 10

3. Example 1B: Solving Inequalities with Variables on Both Sides
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
To collect the variable terms on one side, subtract 2m from both sides.
–2m – 2m
2m – 3 <
Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction
2m <
Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 4 5 6

4. Solve the inequality and graph the solutions.
Check It Out! Example 1a
Solve the inequality and graph the solutions.
4x ≥ 7x + 6
4x ≥ 7x + 6
–7x –7x
To collect the variable terms on one side, subtract 7x from both sides.
–3x ≥ 6
x ≤ –2
Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.
–10 –8 –6 –4 –2 2 4 6 8 10

5. Solve the inequality and graph the solutions.
Check It Out! Example 1b
Solve the inequality and graph the solutions.
5t + 1 < –2t – 6
5t + 1 < –2t – 6
+2t t
7t + 1 < –6
To collect the variable terms on one side, add 2t to both sides.
Since 1 is added to 7t, subtract 1 from both sides to undo the addition.
– 1 < –1
7t < –7
Since t is multiplied by 7, divide both sides by 7 to undo the multiplication.
7t < –7
t < –1 –5 –4 –3 –2 –1 1 2 3 4 5

6. Example 3A: Simplify Each Side Before Solving
Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
Distribute 2 on the left side of the inequality.
2(k – 3) > 3 + 3k
2k + 2(–3) > 3 + 3k
2k – 6 > 3 + 3k
To collect the variable terms, subtract 2k from both sides.
–2k – 2k
–6 > 3 + k
Since 3 is added to k, subtract 3 from both sides to undo the addition.
–3 –3
–9 > k

7. Example 3A Continued
–9 > k
–12 –9 –6 –3 3

8. Check It Out! Example 3a
Solve the inequality and graph the solutions.
5(2 – r) ≥ 3(r – 2)
Distribute 5 on the left side of the inequality and distribute 3 on the right side of the inequality.
5(2 – r) ≥ 3(r – 2)
5(2) – 5(r) ≥ 3(r) + 3(–2)
10 – 5r ≥ 3r – 6
Since 6 is subtracted from 3r, add 6 to both sides to undo the subtraction.
16 − 5r ≥ 3r
Since 5r is subtracted from 16 add 5r to both sides to undo the subtraction.
+ 5r +5r
≥ 8r

9. Check It Out! Example 3a Continued
16 ≥ 8r
Since r is multiplied by 8, divide both sides by 8 to undo the multiplication.
2 ≥ r –6 –2 2 –4 4

10. Check It Out! Example 3b
Solve the inequality and graph the solutions.
0.5x – x < 0.3x + 6
2.4x – 0.3 < 0.3x + 6
Simplify.
2.4x – 0.3 < 0.3x + 6
Since 0.3 is subtracted from 2.4x, add 0.3 to both sides.
2.4x < 0.3x + 6.3
Since 0.3x is added to 6.3, subtract 0.3x from both sides.
–0.3x –0.3x
2.1x <
Since x is multiplied by 2.1, divide both sides by 2.1.
x < 3

11. Check It Out! Example 3b Continued–5 –4 –3 –2 –1 1 2 3 4 5

12. Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. t < 5t + 24
t > –6

13. Lesson Quiz: Part I
Solve each inequality and graph the solutions.
3. 4b + 4(1 – b) > b – 9
b < 13