Holt McDougal Algebra 1 Solving Inequalities with Variables on Both Sides 4m – 3 < 2m + 6 To collect the variable terms on one side, subtract 2m from both

Holt McDougal Algebra 1

Solving Inequalities with Variables on Both Sides

4m – 3 < 2m + 6To collect the variable terms on one

side, subtract 2m from both sides.–2m – 2m

2m – 3 < + 6

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

+ 3 + 3

2m < 9 Since m is multiplied by 2, divide both sides by 2 to undo the multiplication.

4 5 6

Example 1: Solving Inequalities with Variables on Both Sides

Solve the inequality and graph the solutions.

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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.

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Example 2 Continued

312 + 12 • w < 36 • w

312 + 12w < 36w– 12w –12w

312 < 24w

13 < w

The Home Cleaning Company is less expensive for houses with more than 13 windows.

To collect the variable terms, subtract 12w from both sides.

Since w is multiplied by 24, divide both sides by 24 to undo the multiplication.

HomeCleaningCompany

siding charge

plus$12 per window

# of windows

is lessthan

PowerClean

cost per window

# ofwindows.times

times

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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

plus

$0.10per

flyer

is lessthan

# of flyers.

A-PlusAdvertisingfee of $24

Print and

More’s cost

per flyer

# of flyers

times times

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Check It Out! Example 3 Continued

24 + 0.10f < 0.25f

–0.10f –0.10f

24 < 0.15f

160 < f

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

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

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

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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.

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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.

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2(3y – 2) – 4 ≥ 3(2y + 7)

Solve the inequality.

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

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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Documents

Holt McDougal Algebra 1 Solving Inequalities with Variables on Both Sides 4m – 3 < 2m + 6 To collect the variable terms on one side, subtract 2m from both