1. SOLVING EQUATIONS, INEQUALITIES, AND ALGEBRAIC PROPORTIONS
Lesson 23

2. WARM UP
Combine like terms
2(x + 7) – 4(8x + 2)
3x(2 + 7x) + 5(x – 5)

3. WARM UP Combine like terms 2(x + 7) – 4(8x + 2) 3x(2 + 7x) + 5(x – 5)

4. SOLVING EQUATIONS
The goal is to isolate the variable (get the variable on one side alone)
-6 + 2y = -2
-6 + 2y + 6 = Add 6 to both sides
2y = 4
2y/2 = 4/2 Divide both sides by 2
y = 2

5. EXAMPLE 1
x + 4(x – 2) = 28 – x
A. x = -6
B. x = -3
C. x = 3
D. x = 6

6. EXAMPLE 1- SOLUTION
x + 4(x – 2) = 28 – x Distribute x + 4x – 8 = 28 – x Combine like terms 5x – 8 = 28 – x Add x to both sides 5x – 8 + x = 28 – x + x 6x – 8 = 28 Add 8 to both sides 6x – = x = 36 Divide both sides by 6 6x/6 = 36/6 x = 6

7. EXAMPLE 2

8. EXAMPLE 2- SOLUTION
Notice that the first equation in each selection is the same.
7x – 4 = 24
Solve this equations, and then either plug your answer into the other equations, or solve to find the pair that matches.

9. EXAMPLE 2- SOLUTION CONTINUED
7x – 4 = 24 Add 4 to both sides 7x – = x = 28 Divide both sides by 7 7x/7 = 28/7 x = 4

10. EXAMPLE 2- SOLUTION CONTINUED
x = 4 in the first equation.
3(4) + 6 = 9 – 4
= 9 – 4
18 = 5
No, answer is not A
4 + 3(4 + 2) = 2(3 - 4)
4 + 3(6) = 2(-1)
No, answer is not B
6(4 +2) = 9(4)
6(6) = 36
36 – 36
Yes, answer is C.

11. INEQUALITIES Solve these the same as equations.
The only difference is when you multiply or divide both sides by a negative be sure to switch the direction of the inequality.

12. EXAMPLE 3 What is the solution to the inequality below?
4x < 2(x – 5)
A. x < -5
B. x < 5
C. x > -5
D. x > 5

13. EXAMPLE 3- SOLUTION
4x < 2(x – 5) Distribute 4x < 2x – 10 Subtract 2x on both sides 4x – 2x < 2x – x 2x <- 10 Divide both sides by 2 2x/2 < -10/2 x < -5

14. EXAMPLE 4
8(x - 3) ≥ 7x + 2
A. x ≥ -26
B. x ≥ -24
C. x ≥ 24
D. x ≥ 26

15. EXAMPLE 4- SOLUTION
8(x - 3) ≥ 7x + 2 Distribute 8x – 24 ≥ 7x + 2 Subtract 7x on both sides 8x – 24 – 7x ≥ 7x + 2 – 7x x – 24 ≥ 2 Add 24 to both sides x – ≥ x ≥ 26

16. PROPORTIONS
When you have two fractions set equal to each other it is called a proportion.
To solve a proportion cross multiply and set each product equal to each other.

17. EXAMPLE 5

18. EXAMPLE 5- SOLUTION y(10) = 8(y + 4) 10y = 8y + 32

19. EXAMPLE 6

20. EXAMPLE 6- SOLUTION 4(x + 5) = (x – 3)5 4x + 20 = 5x – 15 -x = -35