Solving Inequalities — Practice
Solve them exactly like equations — with one extra rule that catches almost everyone.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Integer — A whole number, positive, negative, or zero.
- Inequality — A statement that one expression is less than or greater than another.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- Range — The set of outputs that actually occur.
- Number line — A line with numbers increasing to the right.
Problem 1
Solve x + 5 > 12. Enter your answer like x>7.
Worked answer
x>7
x > 7
Problem 2
Solve 3x ≤ 21. Enter your answer like x<=7.
Worked answer
x<=7
x ≤ 7
Problem 3
Solve x - 4 ≥ -1. Enter your answer like x>=3.
Worked answer
x>=3
x ≥ 3
Problem 4
On a number line, which dot do you use for x > 5?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
An open (hollow) dot
x > 5 does not include 5 itself, so the dot is open.
Problem 5
Solve x2 > 6. Enter your answer like x>12.
Worked answer
x>12
x > 12
Problem 6
Solve 3x + 4 < 19. Enter your answer like x<5.
Worked answer
x<5
3x < 15
x < 5
Problem 7
Solve -4x + 3 ≥ 19. Enter your answer like x<=-4.
Worked answer
x<=-4
-4x ≥ 16
Divide by -4 and flip: x ≤ -4
Check x = -5: 20 + 3 = 23 ≥ 19 ✓
Problem 8
Solve 5 - 2x > 11. Enter your answer like x<-3.
Worked answer
x<-3
-2x > 6
Divide by -2 and flip: x < -3
Problem 9
Solve 2(x - 3) ≤ 8. Enter your answer like x<=7.
Worked answer
x<=7
x - 3 ≤ 4
x ≤ 7
Problem 10
Ava has $85. She buys a $28 jacket, then T-shirts at $14 each. At most how many T-shirts?
Worked answer
4
14t ≤ 57
t ≤ 4.07…
She can buy 4 shirts.
Problem 11
Which dot do you use for x ≥ 3?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
A closed (filled) dot
3 itself is a solution, so the dot is closed.
Problem 12
Solve 7x - 2 > 4x + 13. Enter your answer like x>5.
Worked answer
x>5
3x - 2 > 13
3x > 15
x > 5
Problem 13
How many integers satisfy -7 ≤ 2x - 1 < 9?
Worked answer
8
Add 1 everywhere: -6 ≤ 2x < 10
Divide by 2: -3 ≤ x < 5
Integers: -3, -2, -1, 0, 1, 2, 3, 4
That is 8 values — note 5 is excluded.
Problem 14
Solve -3(x - 2) < 12. Enter your answer like x>-2.
Worked answer
x>-2
-3x + 6 < 12
-3x < 6
Divide by -3 and flip: x > -2
Problem 15
Priya scored 76, 84 and 79 on three tests. What is the lowest fourth score that gives an average of at least 80?
Worked answer
81
Needed total: 80 × 4 = 320
Current total: 76 + 84 + 79 = 239
320 - 239 = 81
Problem 16
Solve x-5 ≥ 3. Enter your answer like x<=-15.
Worked answer
x<=-15
Multiply both sides by -5 and flip: x ≤ -15
Check x = -20: -20-5 = 4 ≥ 3 ✓