1. Solving Inequalities & Problem Solving with Inequalities

2. Review Inequalities: Less than < x<5 Greater than > x>-2
Less than or equal to ≤ x ≤3
Greater than or equal to ≥ x ≥0

3. To graph Inequalities:
Use an open circle for < or >
then draw an arrow and shade the line in for which direction it will go.
Use a closed circle for ≤ or ≥
then draw an arrow and shade shade the line in for which direction it will go.
Example:
x< 4 1 2 3 4 5 6

4. write and graph each of the following inequalities.
The summer temperature T, in Phoenix is at least 80 degrees.
The average snow fall is less than one inch.
(*use less than but not or equal to)
The class average is 85% and higher
(*use greater than or equal to)
(*use greater than or equal to)
T ≥ 80 degrees 77 78 79 80 81 82
S < 1
A ≥ 85% 82 83 84 85 86 87

5. Tips to Solving Linear Inequalities:
Solve just like an equation
Get the variable alone on one side
Remember if you multiply or divide by a negative number to get the variable alone, reverse (change) the inequality symbol!
-2x < 8
x > -4

6. Example: 2x + 3 > 4 As an equation 2x + 3 = 4 -3 -3 2x = 1 2 2 x =
2x = 1
x =
As an inequality
2x + 3 > 4
2x > 1
x >
Notice the x value is the same. Now graph your answer. 1 2 3

7. x – 8 < -10 -3x + 4 > 19 -3x > 15 x < -2 x < -5
-3x + 4 > 19
-3x > 15
*flip the > to a < because divided by -3
x < -5

8. Practice: solve and graph
x + 3 ≤ 2(x-4)
x + 3 ≤ 2x – 8
-x x
3 ≤ x –
11 ≤ x
Tip: Same meaning as x > 11 and usually easier to decide which way the arrow points when you write the variable on the left

9. In Owego, NY, the temperature in January may not exceed 0 degrees C
In Owego, NY, the temperature in January may not exceed 0 degrees C. Write an inequality that describes temperature T for the month and graph it.
T ≤ 0

10. Problem Solving With Inequalities
You have $32.14 in your pocket that you want to spend on a new pair of shoes. If sales tax is 6%, what’s the most your shoes can cost before tax?
Label what you know:
You have = $32.14
Sales tax = .06
“x” cost of shoes
Write an inequality.
x + x(.06) ≤ 32.14
Solve
1.06x ≤ 32.14
x ≤ 30.32
Answer the question:
The shoes can cost up to $30.32 and you would have enough money to pay for them.

11. The last problem to solve together:
You need some new CD’s to copy your photos on to send to your grandma in St. Louis. You found a great deal where you can get CD’s for $3.00 each plus a flat shipping fee of $ If you have $25.00 to spend, how many CD’s can you order?
Label what you know:
CDs= $3
Shipping = $2
You have = $25
“x” number of CDs
Write an inequality.
3x + 2 ≤ 25
Solve
3x ≤ 23
x ≤ 7 2/3
Answer the question:
You can order up to 7 CDs and have enough money to pay for them.