–2 m – 2 m. 2 m – 3 < + 6. + 3 + 3. 2 m < 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 < 2 m + 6.
2m – 3 < + 6
+ 3 + 3
2m < 9
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.
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.
cost per window
# of windows
$12 per window
– 12w –12w
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.
fee of $24
# of flyers.
# of flyers
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
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.
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.
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.
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.