Solving and Graphing Inequalities

Show your work for each problem.

1.
Which symbol means 'greater than or equal to'?
1. a) $\leq$
2. b) $\geq$
3. c) $<$
4. d) $>$
Answer: $\geq$
$\geq$ means 'greater than or equal to'. The line underneath adds the 'equal to' part. $>$ alone means only 'greater than'.
2.
Solve: $x + 4 > 9$
1. a) $x < 5$
2. b) $x > 5$
3. c) $x > 13$
4. d) $x < 13$
Answer: $x > 5$
Subtract 4 from both sides: $x + 4 - 4 > 9 - 4$ , which gives $x > 5$ . The inequality sign stays the same because we only subtracted.
3.
Solve for $x$ : $x - 3 \leq 7$ . Write the solution in the form 'x <= number'.
Answer: x <= 10
Add 3 to both sides: $x - 3 + 3 \leq 7 + 3$ , so $x \leq 10$ .
4.
Solve: $5x < 25$ . Write your answer as 'x < number'.
Answer: x < 5
Divide both sides by 5: $\frac{5x}{5} < \frac{25}{5}$ , so $x < 5$ . Since 5 is positive, the inequality sign stays the same.
5.
How do you graph $x > 3$ on a number line?
1. a)Closed circle at 3, shade left
2. b)Open circle at 3, shade right
3. c)Closed circle at 3, shade right
4. d)Open circle at 3, shade left
Answer: Open circle at 3, shade right
Since $>$ means strictly greater (not equal), use an open circle at 3 to show 3 is not included. Shade right for 'greater than' since larger numbers are to the right.
6.
Solve: $-3x > 12$
1. a) $x > -4$
2. b) $x < -4$
3. c) $x > 4$
4. d) $x < 4$
Answer: $x < -4$
Divide both sides by $-3$ : $\frac{-3x}{-3} > \frac{12}{-3}$ . Since we divide by a negative number, the sign flips: $x < -4$ .
7.
Solve: $-2x \leq 10$ . Write your answer as 'x >= number' or 'x <= number'.
Answer: x >= -5
Divide both sides by $-2$ : $\frac{-2x}{-2} \leq \frac{10}{-2}$ . Flip the sign when dividing by negative: $x \geq -5$ .
8.
Solve the inequality: $2x + 5 > 13$
Answer: x > 4
1. Subtract 5 from both sides. What do you get? 2x > 8
2. Divide both sides by 2. What is the final answer? x > 4
9.
Solve the inequality: $3x - 7 \leq 11$
Answer: x <= 6
1. Add 7 to both sides. What do you get? 3x <= 18
2. Divide both sides by 3. What is the final answer? x <= 6
10.
Solve the inequality: $4 - 2x \geq 10$
Answer: x <= -3
1. Subtract 4 from both sides. What do you get? -2x >= 6
2. Divide both sides by $-2$ . Remember to flip the sign! What is the answer? x <= -3
11.
Solve: $-5x + 3 < -17$
1. a) $x < 4$
2. b) $x > 4$
3. c) $x < -4$
4. d) $x > -4$
Answer: $x > 4$
Subtract 3: $-5x < -20$ . Divide by $-5$ and flip: $x > 4$ .
12.
Solve: $\frac{x}{4} + 2 > 5$ . Write your answer as 'x > number'.
Answer: x > 12
Subtract 2: $\frac{x}{4} > 3$ . Multiply by 4: $x > 12$ . No sign flip since 4 is positive.
13.
Solve the inequality: $7 - 4x > 2x + 1$
Answer: x < 1
1. Subtract $2x$ from both sides. What do you get? 7 - 6x > 1
2. Subtract 7 from both sides. What do you get? -6x > -6
3. Divide by $-6$ and flip the sign. What is the final answer? x < 1
14.
A gym membership costs 15 dollars per month plus a 30 dollar enrollment fee. If you have 120 dollars, for how many full months can you be a member? Set up and solve an inequality.
Answer: 6
1. Write the inequality. If $m$ is the number of months, what is the total cost? 15m + 30 <= 120
2. Subtract 30 from both sides. What do you get? 15m <= 90
3. Divide both sides by 15. What is $m$ ? m <= 6
4. How many full months can you be a member? 6
15.
Solve: $\frac{2x - 3}{5} \geq 1$
Answer: x >= 4
1. Multiply both sides by 5. What do you get? 2x - 3 >= 5
2. Add 3 to both sides. What do you get? 2x >= 8
3. Divide both sides by 2. What is the final answer? x >= 4