To solve a quadratic inequality, we write our inequality as an equality and solve. The solutions to the equation give us boundaries that allow us to set up intervals on the number line. We can then test values in each interval and determine our solution set. Lastly, our boundaries are included for non-strict inequalities and not included for strict inequalities.

Test Objectives
• Demonstrate the ability to solve a quadratic equation
• Demonstrate the ability to solve a quadratic inequality
• Demonstrate the ability to write an inequality solution in interval notation
• Demonstrate the ability to graph an interval

#1:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}2x^2 - 6x + 2 > -2$$

$$b)\hspace{.2em}6x^2 - 10x + 2 < 6$$

#2:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}6x^2 ≥ 5x^2 + 99 + 2x$$

$$b)\hspace{.2em}6x^2 + 17x - 8 ≤ 9x$$

#3:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}6x^2 - 8x + 1 > -x$$

$$b)\hspace{.2em}5x^2 + 6x < 11$$

#4:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}x^2 - 17x + 5 ≥ -11x$$

$$b)\hspace{.2em}{-}6x^2 - x ≤ 12 - 7x^2$$

#5:

Instructions: solve each inequality, write in interval notation, graph.

$$a)\hspace{.2em}5x^2 - 20x - 32 ≤ -8x$$

$$b)\hspace{.2em}x^2 - 5x ≥ 143 - 3x$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x < 1 \hspace{.2em}or \hspace{.2em}x > 2$$ $$(-\infty, 1) ∪ (2, \infty)$$

$$b)\hspace{.2em}{-}\frac{1}{3}< x < 2$$ $$\left(-\frac{1}{3}, 2 \right)$$

#2:

Solutions:

$$a)\hspace{.2em}x ≤ -9 \hspace{.2em}or \hspace{.2em}x ≥ 11$$ $$(-\infty, -9] ∪ [11, \infty)$$

$$b)\hspace{.2em}{-}2 ≤ x ≤ \frac{2}{3}$$ $$\left[-2, \frac{2}{3}\right]$$

#3:

Solutions:

$$a)\hspace{.2em}x < \frac{1}{6}\hspace{.2em}or \hspace{.2em}x > 1$$ $$\left(-\infty, \frac{1}{6}\right) ∪ (1, \infty)$$

$$b)\hspace{.2em}{-}\frac{11}{5}< x < 1$$ $$\left(-\frac{11}{5}, 1\right)$$

#4:

Solutions:

$$a)\hspace{.2em}x ≤ 1 \hspace{.2em}or \hspace{.2em}x ≥ 5$$ $$(-\infty, 1] ∪ [5, \infty)$$

$$b)\hspace{.2em}{-}3 ≤ x ≤ 4$$ $$[-3, 4]$$

#5:

Solutions:

$$a)\hspace{.2em}{-}\frac{8}{5}≤ x ≤ 4$$ $$\left[-\frac{8}{5}, 4\right]$$

$$b)\hspace{.2em}x ≤ -11 \hspace{.2em}or \hspace{.2em}x ≥ 13$$ $$(-\infty, -11] ∪ [13, \infty)$$