Rational Inequality: Solving Algebraically Example: Solve Find the boundary points. a) Change the inequality to an equation, and solve …

Rational Inequality: Solving Algebraically Example: Solve b) Find the restricted values. These are the values that will make the denominator zero … Here the boundary points are - 2 and 5.

Rational Inequality: Solving Algebraically 2) Put the boundary points on a number line. - 2 5 The two boundary points divide the line up into three sections, called intervals. Slide 2

Rational Inequality: Solving Algebraically - 2 5 Try a test number from each interval in the original inequality to determine if it satisfies the inequality. If it does satisfy the inequality, then all values in the interval will satisfy the inequality. If the test number does not satisfy the inequality, then no value in the interval will satisfy the inequality. Slide 2

Rational Inequality: Solving Algebraically - 2 5 no For example, for the right interval choose x = 10 and the inequality becomes a false statement. Write no on the interval … Slide 2

Rational Inequality: Solving Algebraically Testing the other intervals leads to … x = - 10 x = 0 x = 10 - 2 5 ineq. is false ineq. is true ineq. is false no yes 4) The solution set is therefore the interval (- 2, 5). Slide 3

Rational Inequality: Solving Algebraically Note: had the original inequality included the equals sign … all other work would have been the same, except how we write the answer. With the inclusion of the equals, the boundary points are included, and thus we use brackets in the interval notation. Slide 3

Rational Inequality: Solving Algebraically Written with the brackets, the solution would look like But be careful here. Since – 2 would lead to a zero in the denominator of the original expression, we must exclude it. The final correct result would be… Slide 3

Rational Inequality: Solving Algebraically Try to solve The solution set is (- , 0)  (3, ). Note: If the above inequality were instead of the solution set would be (- , 0)  [3, ). Note, 0 would not be included in the solution set because it makes the inequality undefined (zero in the denominator). Slide 4

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