Inequalities

1. Solve the following inequalities. Write the solution in interval notation. Then graph the solution.

(a) \(\displaystyle{ 2(x + 1) - 2 \leq 3(2 - x) + 9 } \)    solution

(b) \(\displaystyle{ 2[x - 3(4x + 1)] \geq 37 - (2x + 3)} \)    solution

2. Solve the following inequalities. Write the solution in interval notation. Then graph the solution.

(a)  \(\displaystyle{ -1 < \frac{2k + 6}{4} < 11 }\)    solution

(b)  \(\displaystyle{ \frac{2x + 1}{3} < \frac{x - 1}{2} + \frac{1}{6} }\)    solution

3. Solve each inequality. Write the solution in interval notation. Then graph the solution.

(a)  \(\displaystyle{3x - 2 > 7}\)  or  \(\displaystyle{2(1 - x) > 1} \)    solution

(b)  \(\displaystyle{\frac{x + 2}{2} < \frac{x}{3} + 1} \)  or  \(\displaystyle{\frac{x - 1}{3} > \frac{x + 1}{5}} \)    solution

4. Solve each inequality. Write the solution in interval notation. Then graph the solution.

(a)  \(\displaystyle{6 - x \leq 3x + 10} \)  and  \(\displaystyle{7x - 14  \leq 3x + 14} \)    solution

(b)  \(\displaystyle{5(x + 2) + 7 < 2} \)  and  \(\displaystyle{2(5 - 3x) + 1 < 17} \)    solution

5. Solve each inequality. Write the solution in interval notation. Then graph the solution.

(a)  \(\displaystyle{ 9 \leq  x + 2 \leq 17 }\)    solution

(b)  \(\displaystyle{ -2 \leq 3x + 1 < 7 }\)    solution

6. Find \(\displaystyle{a} \) and \(\displaystyle{b} \).

(a)  If \(\displaystyle{-1 < x < 1} \), then \(\displaystyle{a < 2 - x < b} \).    solution

(b)  If \(\displaystyle{1 < x < 5} \), then \(\displaystyle{a < 2x + 3 < b} \).    solution

7. Solve the following inequalities. Write the answer in interval notation.

(a) \(\displaystyle{ x^2 - 4x \geq 12 }\)  solution

(b) \(\displaystyle{ \frac{x + 3}{x - 5} < 0 }\)   solution

(c) \(\displaystyle{ \frac{x + 3}{x + 6} \geq 5 }\)   solution

(d) \(\displaystyle{ \frac{(x + 3)(x - 5)}{x + 6} \leq 0 }\)   solution

(e) \(\displaystyle{ (2-x)^2 \left(x-\frac{5}{2}\right) < 0}\)   solution