1. Lesson Objective: I will be able to …
Solve inequalities in one variable that contain
variable terms on both sides
Language Objective: I will be able to …
Read, write, and listen about vocabulary, key
concepts, and examples

2. Example 1: Solving Inequalities with Variables on Both Sides
Page 32
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

3. Solve the inequality and graph the solutions.
Your Turn 1
Page 32
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

4. Example 2: Business Application
Page 33
The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for each window. Power Clean charges $36 per window, and the price includes power-washing the siding.
How many windows must a house have to make the total cost from The Home Cleaning Company less expensive than Power Clean?
Let w be the number of windows.
Home
Cleaning
Company
siding charge
plus
$12 per window
# of windows
is less
than
Power
Clean
cost per window
# of
windows.
times
• w < • w

5. Example 2 Continued
w < 36w
– 12w –12w
To collect the variable terms, subtract 12w from both sides.
312 < 24w
Since w is multiplied by 24, divide both sides by 24 to undo the multiplication.
13 < w
The Home Cleaning Company is less expensive for houses with more than 13 windows.

6. Example 3: Simplify Each Side Before Solving
Page 34
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
–12 –9 –6 –3 3

7. Your Turn 3 Solve the inequality and graph the solutions.
Page 34
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
Since r is multiplied by 8, divide both sides by 8 to undo the multiplication.
2 ≥ r –6 –2 2 –4 4

8. Page 31

9. Example 4: Identities and Contradictions
Page 34
Solve the inequality.
2x – 7 ≤ 5 + 2x
2x – 7 ≤ 5 + 2x
–2x –2x
Subtract 2x from both sides.
–7 ≤ 5 
True statement.
The inequality 2x − 7 ≤ 5 + 2x is an identity. All values of x make the inequality true. Therefore, all real numbers are solutions.

10. Example 5: Identities and Contradictions
Page 35
Solve the inequality.
2(3y – 2) – 4 ≥ 3(2y + 7)
Distribute 2 on the left side and 3 on the right side.
2(3y – 2) – 4 ≥ 3(2y + 7)
2(3y) – 2(2) – 4 ≥ 3(2y) + 3(7)
6y – 4 – 4 ≥ 6y + 21
6y – 8 ≥ 6y + 21
–6y –6y
Subtract 6y from both sides. 
–8 ≥ 21
False statement.
No values of y make the inequality true. There are no solutions.

11.  Your Turn 5 Solve the inequality. x – 2 < x + 1 x – 2 < x + 1
Page 35
Solve the inequality.
x – 2 < x + 1
x – 2 < x + 1
–x –x
Subtract x from both sides.
–2 < 1 
True statement.
All values of x make the inequality true.
All real numbers are solutions.

12. Homework
Assignment #6
3-5 #4-6, 14, 16