1. Warm Up Solve each equation. 1. 2x = 7x + 15 2. x = –3
3y – 21 = 4 – 2y
y = 5
3. 2(3z + 1) = –2(z + 3)
z = –1
4. 3(p – 1) = 3p + 2
no solution
5. Solve and graph 5(2 – b) > 52.
b < –3 –5 –3 –2 –1 –4 –6

2. Additional 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.

3. Additional Example 1A: Continued
Solve the inequality and graph the solutions.
y ≤ 4y + 18
The solution set is {y:y ≥ –6}.
–6 ≤ y (or y  –6)
–10 –8 –6 –4 –2 2 4 6 8 10

4. Additional 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.

5. Additional Example 1B Continued
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
The solution set is {m:m }. 4 5 6

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

7. Check It Out! Example 1a Continued
Solve the inequality and graph the solutions. Check your answer.
4x ≥ 7x + 6
Check
Check the endpoint, –2. 
– –14 + 6
4(–2) 7(–2) + 6
– –8
4x = 7x + 6 
Check a number less
than –2.
–12 ≥ –15
4x ≥ 7x + 6
4(–3) ≥ 7(–3) + 6
–12 ≥ –21 + 6

8. Solve the inequality and graph the solutions. Check your answer.
Check It Out! Example 1b
Solve the inequality and graph the solutions. Check your answer.
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
The solution set is {t:t < –1}. –5 –4 –3 –2 –1 1 2 3 4 5

9. Additional 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

10. Additional Example 3A Continued
Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
–9 > k
The solution set is {k:k < –9}.
–12 –9 –6 –3 3

11. Check It Out! Example 3a
Solve the inequality and graph the solutions. Check your answer.
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

12. Check It Out! Example 3a Continued
Solve the inequality and graph the solutions. Check your answer.
16 ≥ 8r
Since r is multiplied by 8, divide both sides by 8 to undo the multiplication.
2 ≥ r
The solution set is {r:r ≤ 2}. –6 –2 2 –4 4

13. Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. t < 5t + 24
t > –6
2. 5x – 9 ≤ 4.1x – 81
x ≤ –80
3. 4b + 4(1 – b) > b – 9
b < 13

14. Lesson Quiz: Part II
4. Rick bought a photo printer and supplies for $186.90, which will allow him to print photos for $0.29 each. A photo store charges $0.55 to print each photo. How many photos must Rick print before his total cost is less than getting prints made at the photo store?
Rick must print more than 718 photos.

15. Lesson Quiz: Part III
Solve each inequality.
5. 2y – 2 ≥ 2(y + 7) ø
6. 2(–6r – 5) < –3(4r + 2)
all real numbers