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–2 m – 2 m. 2 m – 3 &lt; + 6. + 3 + 3. 2 m &lt; 9. 4. 5. 6. 5.2. Example 1: Solving Inequalities with Variables on Both Sides. Solve the inequality and graph the solutions. 4 m – 3 &lt; 2 m + 6.

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slide1

–2m –2m

2m – 3 < + 6

+ 3 + 3

2m < 9

4

5

6

5.2

Example 1: 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.

Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction

Since m is multiplied by 2, divide both sides by 2 to undo the multiplication.

slide2

5.2

Example 2: Business Application

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.

slide3

Home

Cleaning

Company

siding charge

Power

Clean

cost per window

is less

than

# of

windows.

# of windows

times

$12 per window

plus

times

– 12w –12w

5.2

Example 2 Continued

312 + 12 • w < 36 • w

312 + 12w < 36w

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.

slide4

$0.10

per flyer

Print and

More’s cost

per flyer

A-Plus

Advertising

fee of $24

# of flyers.

# of flyers

is less

than

times

times

plus

5.2

Check It Out! Example 3

A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More?

Let f represent the number of flyers printed.

24 + 0.10 • f < 0.25 • f

slide5

–0.10f –0.10f

5.2

Check It Out! Example 3 Continued

24 + 0.10f < 0.25f

To collect the variable terms, subtract 0.10f from both sides.

24 < 0.15f

Since f is multiplied by 0.15, divide both sides by 0.15 to undo the multiplication.

160 < f

More than 160 flyers must be delivered to make A-Plus Advertising the lower cost company.

slide6

5.2

Some inequalities are true no matter what value is substituted for the variable. For these inequalities, all real numbers are solutions.

Some inequalities are false no matter what value is substituted for the variable. These inequalities have no solutions.

If both sides of an inequality are fully simplified and the same variable term appears on both sides, then the inequality has all real numbers as solutions or it has no solutions. Look at the other terms in the inequality to decide which is the case.

slide7

5.2

Additional Example 4: All Real Numbers as Solutions or No Solutions

Solve the inequality.

2x – 7 ≤ 5 + 2x

The same variable term (2x) appears on both sides. Look at the other terms.

For any number 2x, subtracting 7 will always result in a lower number than adding 5.

All values of x make the inequality true.

All real numbers are solutions.

slide8

5.2

Additional Example 5: All Real Numbers as Solutions or No Solutions

Solve the inequality.

2(3y –2) – 4 ≥ 3(2y + 7)

Distribute 2 on the left side and 3 on the right side and combine like terms.

6y –8 ≥ 6y + 21

The same variable term (6y) appears on both sides. Look at the other terms.

For any number 6y, subtracting 8 will never result in a higher number than adding 21.

No values of y make the inequality true.

There are no solutions.

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