334 CHAPTER 6 Algebra, Graphs, and Functions Recall from Section 6.1 that when an equation contains fractions, we generally begin by multiplying both sides of the equation by the lowest common denominator (LCD) of the fractions. This step eliminates the fractions from the equation. We can do a similar step with inequalities as seen in our next example. Example 8 An Inequality that Contains Fractions Solve the inequality x x , 3 4 3 8 1 2 1 8 + ≤ − and graph the solution set on the number line. Solution Our first step is to multiply both sides of the inequality by least common denominator, 8. This will eliminate the fractions from the inequality. + ≤ − ⎛ + ⎝ ⎞ ⎠ ≤ ⎛ − ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ ≤ ⎛ ⎝ ⎞ ⎠ − ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ≤ ⎛ ⎝⎜ ⎞ ⎠⎟ − ⎛ ⎝⎜ ⎞ ⎠⎟ + ≤ − + − ≤ − − + ≤ − + − ≤ − − ≤ − ≤ − ≤ − x x x x x x x x x x x x x x x x x x x 3 4 3 8 1 2 1 8 8 3 4 3 8 8 1 2 1 8 8 3 4 8 3 8 8 1 2 8 1 8 8 3 4 8 3 8 8 1 2 8 1 8 6 3 4 1 6 3 4 4 1 4 2 3 1 2 3 3 1 3 2 4 2 2 4 2 2 2 1 4 1 Multiply both sides by LCD Distributive property Simplify Subtract x4 from both sides Subtract 3 from both sides Divide both sides by 2 The solution set is graphed on the number line as follows. x –2 27 26 25 24 23 22 21 0 1 2 3 7 Now try Exercise 33 MATHEMATICS TODAY What a Bore In the production of machine parts, engineers must allow a certain tolerance in the way parts fit together. For example, the boring machines that grind cylindrical openings in an automobile’s engine block must create a cylinder that allows the piston to move freely up and down, but still fit tightly enough to ensure that compression and combustion are complete. The allowable tolerance between parts can be expressed as an inequality. For example, the diameter, or bore , of a cylinder may need to be no less than 3.383 in. and no greater than 3.387 in. We can represent the allowable tolerance as t 3.383 3.387. ≤ ≤ Why This Is Important Inequalities can be used to model many applications including the one described above. Understanding compound inequalities like those presented in Examples 9, 10, and 11 is an important skill for many employees working in a variety of industries. Solving Compound Inequalities An inequality of the form a x b < < is called a compound inequality . Consider the compound inequality x 3 2, − < ≤ which means that x 3− < and x 2. ≤ Example 9 A Compound Inequality Graph the solution set of the inequality x 3 2. − < ≤ Solution The solution set consists of all the real numbers greater than 3− and less than or equal to 2. 25 24 23 22 21 0 1 2 3 4 5 23 , x # 2 7 Now try Exercise 35 Colin Hayes/Shutterstock
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