### About Quadratic & Rational Inequalities:

When we are asked to solve quadratic or rational inequalities, we draw on our ability to solve equations, along with our knowledge of inequalities. In each case, we are setting up an equation and using test numbers to determine which intervals satisfy the inequality. The endpoints are always considered separately. We must reject any value that results in a zero denominator.

Test Objectives

- Demonstrate a general understanding of inequalities
- Demonstrate the ability to solve a quadratic inequality
- Demonstrate the ability to solve a rational inequality

#1:

Instructions: Solve each inequality.

a) $$3x^2 + 12x \lt -8 - 2x$$

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#2:

Instructions: Solve each inequality.

a) $$15 + 13r \lt -2r^2$$

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#3:

Instructions: Solve each inequality.

a) $$3x^2 - 20x \le - 25$$

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#4:

Instructions: Solve each inequality.

a) $$\frac{x - 1}{x + 1}\ge - 3$$

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#5:

Instructions: Solve each inequality.

a) $$\frac{(x - 2)^2}{x^2 - 25}\ge 1$$

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Written Solutions:

#1:

Solutions:

a) $$-4 \lt x \lt -\frac{2}{3}$$ $$\left(-4,-\frac{2}{3}\right)$$

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#2:

Solutions:

a) $$-5\lt r \lt -\frac{3}{2}$$ $$\left(-5,-\frac{3}{2}\right)$$

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#3:

Solutions:

a) $$\frac{5}{3}\le x \le 5$$ $$\left[\frac{5}{3},5\right]$$

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#4:

Solutions:

a) $$x \lt -1 \hspace{.25em}or \hspace{.25em}x \ge -\frac{1}{2}$$ $$(-\infty,-1) ∪ \left[-\frac{1}{2},\infty\right)$$

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#5:

Solutions:

a) $$x \lt -5 \hspace{.25em}or \hspace{.25em}5 \lt x \le \frac{29}{4}$$ $$(-\infty,-5) ∪ \left(5,\frac{29}{4}\right]$$