College Algebra

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If the inequality is not written in this form, first rewrite it—as indicated in Step 1.

Slide 32

E.g. 3—A Quadratic Inequality

Solve the inequality

x2 ≤ 5x – 6

First, we move all the terms to the left-hand side x2 – 5x + 6 ≤ 0

Factoring the left side of the inequality, we get (x – 2)(x – 3) ≤ 0

Slide 33

E.g. 3—A Quadratic Inequality

The factors of the left-hand side are x – 2 and x – 3.

These factors are zero when x is 2 and 3, respectively.

Slide 34

E.g. 3—A Quadratic Inequality

As shown, the numbers 2 and 3 divide the real line into three intervals: (-∞, 2), (2, 3), (3, ∞)

Slide 35

E.g. 3—A Quadratic Inequality

The factors x – 2 and x – 3 change sign only at 2 and 3, respectively.

So, they maintain their signs over the length of each interval.

Slide 36

E.g. 3—A Quadratic Inequality

On each interval, we determine the signs of the factors using test values.

We choose a number inside each interval and check the sign of the factors x – 2 and x – 3 at the value selected.

Slide 37

E.g. 3—A Quadratic Inequality

For instance, let’s use the test value x = 1 for the interval (-∞, 2).

Slide 38

E.g. 3—A Quadratic Inequality

Then, substitution in the factors x – 2 and x – 3 gives:

x – 2 = 1 – 2 = –1 < 0

and

x – 3 = 1 – 3 = –2 < 0

So, both factors are negative on this interval.

Notice that we need to check only one test value for each interval because the factors x – 2 and x – 3 do not change sign on any of the three intervals we found.

Slide 39

E.g. 3—A Quadratic Inequality

Using the test values x = 2½ and x = 4 for the intervals (2, 3) and (3, ∞), respectively, we construct the following sign table.

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E.g. 3—A Quadratic Inequality

The final row is obtained from the fact that the expression in the last row is the product of the two factors.

Slide 41

E.g. 3—A Quadratic Inequality

If you prefer, you can represent that information on a real number line—as in this sign diagram.