Ch. 6 Equations and Inequalities

1. Ch. 6Equations and Inequalities 6.2 Linear Equations in One Variable and Proportions

2. Objectives • Solve linear equations. • Solve linear equations containing fractions. • Solve proportions. • Solve problems using proportions. • Identify equations with no solution or infinitely many solutions.

3. Linear Equations • linear equation in one variable x is an equation that can be written in the formax+ b = 0, where a and b are real numbers, and a  0. • Solving an equation in x determining all values of x that result in a true statement when substituted into the equation. • Equivalent equationshave the same solution set. 4x + 12 = 0 and x = 3 are equivalent equations.

4. Addition and Multiplying Properties • The Addition Property of EqualityIf a = b, then a + c = b + c • The Multiplication Property of EqualityIf a = b, then a · c = b · c

5. Using Properties of Equality to Solve Equations

6. Solving Linear Equation Solve and check: 2(x – 4) – 5x = 5. Step 1. Simplify the algebraic expression on each side:2(x– 4) – 5x = 52x– 8 – 5x = 5.3x – 8 = 5 Step 2. Collect variable terms on one side and constants on the other side.3x – 8 + 8 = 5 + 8 3x = 3

7. Solving Linear Equation Step 3. Isolate the variable and solve. 3x = 3 3 3 x =1 Step 4. Check the proposed solution in the original equation by substituting 1 for x. 2(x– 4) – 5x = 5 2(1 – 4) – 5(1) = 5 10 – (5) = 5 5 = 5Because the check results in a true statement, we conclude that the solution set of the given equation is {1}.

8. Application These graphs indicate that persons with a low sense of humor have higher levels of depression. These graphs can be modeled by the following formulas:

9. We are interested in the intensity of a negative life event with an average level of depression of 7/2 for the high humor group.

10. Linear Equation with No Solution Solve: 2x + 6 = 2(x + 4) Solution: 2x + 6 = 2(x + 4) 2x + 6 = 2x + 8 2x + 6 – 2x = 2x + 8 – 2x 6 = 8 The original equation 2x + 6 = 2(x + 4) is equivalent to 6 = 8, which is false for every value of x. The equation has no solution. The solution set is Ø.

11. Linear Equation with Infinitely Many solutions Solve: 4x + 6 = 6(x + 1) – 2x Solution: 4x + 6 = 6(x + 1) – 2x 4x + 6 = 6x + 6 – 2x 4x + 6 = 4x + 6 The original statement is equivalent to the statement 6 = 6, which is true for every value of x. The solution set is the set of all real numbers, expressed as {x|x is a real number}.

12. Proportions • The property tax on a house with an assessed value of $480,000 is $5760. Determine the property tax on a house with an assessed value of $600,000, assuming the same tax rate.5760 x ------------ = ------------ 480,000 600,000

13. 5760 x ------------ = ------------ 480,000 600,000 5760 x ----------- = -------- 48 60 5760 ∙ 60 ------------- = x 48 5 1440 155760 ∙ 60------------ = x 48 12 4 1 7200 = x $7200 = x

14. Example: Changing Recipe Size A chocolate-chip recipe for five dozen cookies requires ¾ cup of sugar. If you want to make eight dozen cookies, how much sugar is needed? Solution: 3 / 4 (cup) x (cup) ----------------- = -------------- 5 (dozen) 8 (dozen) 8 (3/4) 8 3 6---------- = x --- ∙ --- = --- = x 5 5 4 5

15. x 60” 240” 960” Find the height x of the tree, when a 60”-man casts a shadow 240” long.

16. Proportion 60 x ----- = ------- 240 960 60 ∙ 960 ----------- = x x = 240240 x 60” 240” 960”