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

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Example: Solve

Rational Inequality: Solving Algebraically

  • Find the boundary points.

  • a) Change the inequality to an equation, and solve …


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Example: Solve

Rational Inequality: Solving Algebraically

b) Find the restricted values. These are the values that will make the denominator zero …

Here the boundary points are - 2 and 5.


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- 2 5

Rational Inequality: Solving Algebraically

2) Put the boundary points on a number line.

The two boundary points divide the line up into three sections, called intervals.

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- 2 5

Rational Inequality: Solving Algebraically

  • 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.

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- 2 5

For example, for the right interval choose x = 10 and the inequality becomes

a false statement.

Rational Inequality: Solving Algebraically

no

Write no on the interval …

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x = - 10 x = 0 x = 10

ineq. is false ineq. is true ineq. is false

no

no

yes

- 2 5

Rational Inequality: Solving Algebraically

Testing the other intervals leads to …

4) The solution set is therefore the interval (- 2, 5).

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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.

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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…

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Try to solve

Note: If the above inequality were

instead of

the solution set would be (- , 0)  [3, ).

Rational Inequality: Solving Algebraically

The solution set is (- , 0)  (3, ).

Note, 0 would not be included in the solution set because it makes the inequality undefined (zero in the denominator).

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Rational Inequality: Solving Algebraically

END OF PRESENTATION

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