Exercises

1. Determine whether the given value is a solution to the inequality.
   1. \(2k-7 \gt 10\); \(k=9\)
   2. \(20 \lt 3t+8\); \(t=4\)
   3. \(17-3x \geq 26\); \(x=9\)
   4. \(-18 \leq \frac{p}{2} + 20\); \(p=-18\)
2. For each inequality, explain how you would solve it by
   • underlining the first operation you would reverse, and
   • circling the second operation you would reverse.
   1. \(2p-9 \lt 21\)
   2. \(4a + 8 \gt 15\)
   3. \(16 \leq \dfrac{m}{3} + 10\)
3. Consider the inequality \(17-3y \gt 26\). Explain how you would solve this inequality.
4. Solve each inequality. Show your algebraic steps and verify your answers.
   1. \(15x - 9 \gt 21\)
   2. \(3 + \dfrac{k}{2} \geq 11\)
   3. \(33-8w \leq -7\)
   4. \(14 \gt \dfrac{r}{5} + 8\)
   5. \(12 - \dfrac{y}{3} \lt 16\)
5. Graph, on a number line, the solution from each part of question 4.
6. Find the smallest integer value for \(x\) that satisfies each of the following inequalities.
   1. \(6x - 19 \geq -15\)
   2. \(9+5x \gt 3\)
7. Find the greatest integer value for \(x\) that satisfies each of the following inequalities.
   1. \(8 + 3x \lt -6\)
   2. \(10x - 25 \leq -3\)
8. Ammar is joining a tennis club that charges a monthly membership fee of \($15\) and then an admission fee of \($2\) per entry. He has budgeted to spend no more than \($50\) a month on tennis. How many times can Ammar visit the club in a month?
9. A 3D printing studio offters two pricing options for people wanting to use their printers.
   • Option 1: \($0.12\) per gram of filament (printing material) used
   • Option 2: \($5\) per month plus \($0.06\) per gram of filament used
   How much filament (in grams) should you use per month in order to make Option 2 less expensive than Option 1? Round your answer to two decimal places.
10. Fill in each \(\boxed{\phantom{\square}}\) with an inequality sign so that the statement is true.
    1. If \(s \leq t\), then \(-8s+9 \; \boxed{\phantom{\square}} -8t + 9\).
    2. If \(a \gt b\), then \(2a - 7 \; \boxed{\phantom{\square}} \; 2b-7\).
11. 1. Write an inequality that uses the variable \(k\), multiplication, subtraction, and that has \(k \lt 5\) as the solution.
    2. Write an inequality that uses the variable \(k\), division, addition, and that has \(k \lt 5\) as the solution.
12. The lengths of the sides of \(\triangle ABC\) in centimeters are positive integers. \(BC\) is \(7\) cm longer than \(AB\), and \(AC\) is \(4\) cm longer than \(AB\).
    1. Draw \(\triangle ABC\) if \(AB\) is \(10\) cm.
    2. Explain why you cannot draw \(\triangle ABC\) if \(AB\) is \(1\) cm.
    3. Find the smallest possible value for \(AB\) for which you can draw \(\triangle ABC\).
    4. Challenge: Using inequalities, write a general relationship between the side lengths of a triangle.