186 Chapter 2 Linear Equations and Inequalities Section 2.7 Mixture Applications and Uniform Motion Examples Example 1 illustrates a mixture problem. Example 1 How much 80% disinfectant solution should be mixed with 8 L of a 30% disinfectant solution to make a 40% solution? Subtract 0.40x. Subtract 2.4. Divide by 0.40. 0.80x 0.30182 0.401x 82 0.80x 2.4 0.40x 3.2 0.40x 2.4 3.2 0.40x 0.80 x 2 2 L of 80% solution is needed. Examples Example 2 illustrates a uniform motion problem. Example 2 Jack and Diane participate in a bicycle race. Jack rides the first half of the race in 1.5 hr. Diane rides the second half at a rate 5 mph slower than Jack and completes her portion in 2 hr. How fast does each person ride? Distance Rate Time Jack 1.5x x 1.5 Diane 2(x 5) x 5 2 a Distance a b Jack rides distance Diane rides b 1.5x 21x 52 Subtract 2x. Divide by 0.5. 1.5x 2x 10 0.5x 10 x 20 Jack’s speed is x. Jack rides 20 mph. Diane’s speed is x 5, which is 15 mph. Section 2.8 Linear Inequalities Key Concepts A linear inequality in one variable, x, is any relationship in the form: ax b 6 c, ax b 7 c, ax b c, or ax b c, a 0 where . The solution set to an inequality can be expressed as a graph or in set-builder notation or in interval notation. When graphing an inequality or when writing interval notation, a parenthesis, ( or ), is used to denote that an endpoint is not included in a solution set. A square bracket, or , is used to show that an endpoint is included in a solution set. A parenthesis ( or ) is always used with and , respectively. a 6 x 6 b The inequality is used to show that x is greater than a and less than b. That is, x is between a and b. Multiplying or dividing an inequality by a negative quantity requires the direction of the inequality sign to be reversed. Example Example 1 Subtract 6. Simplify. Divide by 2. Reverse the inequality sign. 2x 6 14 2x 6 6 14 6 2x 8 2x 2 x 4 Graph: Set-builder notation: Interval notation: 1, 44 5x 0 x 46 4 8 2 80% 30% 40% Solution Solution Solution Amount of Solution x 8 x 8 Amount of Pure 0.80x 0.30(8) 0.40(x 8) Disinfectant
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