SOLVING EQUATIONS, INEQUALITIES, AND ALGEBRAIC PROPORTIONS Lesson 23

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

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

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

EXAMPLE 1 x + 4(x – 2) = 28 – x A. x = -6 B. x = -3 C. x = 3 D. x = 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 – 8 + 8 = 28 + 8 6x = 36 Divide both sides by 6 6x/6 = 36/6 x = 6

EXAMPLE 2

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.

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

EXAMPLE 2- SOLUTION CONTINUED x = 4 in the first equation. 3(4) + 6 = 9 – 4 12 + 6 = 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.

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.

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

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

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

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 – 24 + 24 ≥ 2 + 24 x ≥ 26

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.

EXAMPLE 5

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

EXAMPLE 6

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