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Section 2.11 : Linear Inequalities

For problems 1 – 10 solve each of the following inequalities. Give the solution in both inequality and interval notations.

1. \(7x + 2\left( {4 - x} \right) < 12 - 3\left( {5 + 6x} \right)\)
2. \(10\left( {3 + w} \right) \ge 9\left( {2 - 4w} \right)\)
3. \(2\left( {4 + 5y} \right) \le 12y - 6\left( {1 - 3y} \right)\)
4. \(2\left( {\displaystyle \frac{1}{3} - \frac{1}{6}z} \right) > \displaystyle \frac{1}{9}z + 4\left( {2 - \frac{7}{{18}}z} \right)\)
5. \(2 \le 2 + 4\left( {3 - x} \right) \le 6\)
6. \( - 4 < 7x + 8 \le 1\)
7. \(\displaystyle \frac{1}{2} < 2\left( {\frac{1}{4} + \frac{1}{8}t} \right) < \frac{3}{4}\)
8. \( - 12 \le 4 - 11m \le 3\)
9. \(\displaystyle 0 \le \frac{3}{7} - \frac{5}{{14}}x < \frac{1}{2}\)
10. \( - 8 < 2\left( {3 + 4x} \right) - 4\left( {1 + 3x} \right) \le 3\)
11. If \( - 7 < x \le 6\) determine a and b for the inequality : \(a < 3x + 8 \le b\)
12. If \( - 3 \le x \le - 1\) determine a and b for the inequality : \(a \le 6 - 2x \le b\)