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Solve for $x$ . $\newline$ (x - 4)(x + 5) < 0 $\newline$ Write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

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Solve quadratic inequalities

Full solution

Q. Solve for

x

.

\newline

(x - 4)(x + 5) < 0

\newline

Write a compound inequality like

1 < x < 3

or like

x < 1

or

x > 3

.

Find Zeros: Find the zeros of the inequality by setting each factor equal to zero. $\newline$ $(x - 4) = 0$ gives $x = 4$ . $\newline$ $(x + 5) = 0$ gives $x = -5$ .
Determine Intervals: Determine the intervals to test based on the zeros. $\newline$ The intervals are $(-\infty, -5$ ), $(-5, 4$ ), and $(4, \infty)$ .
Test $-\infty$ to $-5$ : Test a point from the interval $(-\infty, -5)$ , say $x = -6$ .((-6) - 4)((-6) + 5) = (-10)(-1) > 0, so this interval does not satisfy the inequality.
Test $-5$ to $4$ : Test a point from the interval $(-5, 4)$ , say $x = 0$ .(0 - 4)(0 + 5) = (-4)(5) < 0, so this interval satisfies the inequality.
Test $4$ to $\infty$ : Test a point from the interval $(4, \infty)$ , say $x = 5$ . $\newline$ (5 - 4)(5 + 5) = (1)(10) > 0, so this interval does not satisfy the inequality.
Write Compound Inequality: Write the solution as a compound inequality using the interval that satisfies the inequality. -5 < x < 4.

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