1. EXAMPLE 2 Multiply by the LCD Solve. Check your solution. x – 2 x 1 5 2 + = SOLUTION x – 2 x 1 5 2 + = Multiply by LCD, 5(x – 2). 5(x – 2) x – 2 x 1 5 + 5(x – 2) x – 2 2 =5(x – 2) Multiply and divide out common factors. x – 2 x 5(x – 2) + 5(x – 2) 5 x – 2 2 5(x – 2) = 5x + x – 2 = 10 Simplify. Write original equation.

2. EXAMPLE 2 Multiply by the LCD 6x – 2 = 10 Combine like terms. 6x6x = 12 Add 2 to each side. x = 2 Divide each side by 6. The solution appears to be 2, but the expressions and are undefined when x = 2. So, 2 is an extraneous solution. x – 2 x 2 ANSWER There is no solution.

3. EXAMPLE 3 Factor to find the LCD Solve. Check your solution. 3 x – 7 + 1 8 x 2 – 9x + 14 = SOLUTION Write each denominator in factored form. The LCD is (x – 2)(x – 7). 3 x – 7 + 1 8 (x – 2)(x – 7) = 3 x – 7 + 1 (x – 2)(x – 7) 8 = 3(x – 2)(x – 7) x – 7 + (x – 2)(x – 7) = 8 (x – 2)(x – 7) (x – 2)(x – 7)

4. EXAMPLE 3 Factor to find the LCD 3(x – 2) + (x 2 – 9x + 14) = 8 x 2 – 6x + 8= 8 x 2 – 6x = 0 x(x – 6)= 0 x = 0 or x – 6 = 0 x = 0 or x = 6 ANSWER The solutions are 0 and 6.

5. EXAMPLE 3 Factor to find the LCD CHECK If x = 0: If x = 6: ? 3 0 – 7 + 1 8 0 2 – 9 0 + 14 = 3 6 – 7 + 1 8 6 2 – 9 6 + 14 = ? 4 4 7 = 7 –2–2 = –2–2

6. EXAMPLE 4 Solve a multi-step problem PAINT MIXING You have an 8 pint mixture of paint that is made up of equal amounts of yellow paint and blue paint. To create a certain shade of green, you need a paint mixture that is 80% yellow. How many pints of yellow paint do you need to add to the mixture? SOLUTION Because the amount of yellow paint equals the amount of blue paint, the mixture has 4 pints of yellow paint. Let p represent the number of pints of yellow paint that you need to add.

7. EXAMPLE 4 Solve a multi-step problem STEP 1 Write a verbal model. Then write an equation. STEP 2 Solve the equation. 4 + p 8 + p = 0.8 Write equation.

8. EXAMPLE 4 Solve a multi-step problem 4 + p = 0.8(8 + p) Cross products property 4 + p = 6.4 + 0.8p Distributive property 0.2p = 2.4 Rewrite equation. p = 12 Solve for p. ANSWER You need to add 12 pints of yellow paint. CHECK 4 + p 8 + p = 0.8 Write original equation. 4 + 12 8 + 12 = 0.8 ? Substitute 12 for p.

9. EXAMPLE 4 Solve a multi-step problem = 0.8 ? 16 20 Simplify numerator and denominator. Write fraction as decimal. Solution checks. 0.8 = 0.8

10. GUIDED PRACTICE for Examples 2, 3, and 4 Solve the equation. Check your solution. 3. a a + 4 3 = + 1 – 12 ANSWER a = –10 n n – 11 – 1 22 n 2 – 5n – 66 = 4. ANSWER a = –4:

11. GUIDED PRACTICE for Examples 2, 3, and 4 WHAT IF? In Example 4, suppose you need a paint mixture that is 75% yellow. How many pints of yellow paint do you need to add to the mixture? 5. ANSWER You need to add 8 pints of yellow paint.