Linear Inequalities — Practice
Solve it like an equation, with one extra rule: multiplying or dividing by a negative reverses the sign.
What these questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Inequality — A statement that one expression is less than or greater than another.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
Problem 1
Solve x + 7 > 12.
Worked answer
x > 5
x > 5
No flip: subtracting never changes the direction
Problem 2
Solve 4x ≤ 20.
Worked answer
x <= 5
x ≤ 5
Problem 3
Solve -2x > 8.
Worked answer
x < -4
x < -4
Check x = -5: -2(-5) = 10 > 8 ✓
Problem 4
Solve -x ≤ 3.
Worked answer
x >= -3
x ≥ -3
Problem 5
For the solution x ≥ 4, is the circle at 4 open or filled?
Worked answer
filled
x = 4 is a solution
So the circle is filled
Problem 6
Solve 3x - 5 ≥ 16.
Worked answer
x >= 7
3x ≥ 21
x ≥ 7
Problem 7
Solve 8 - 3x < 23.
Worked answer
x > -5
-3x < 15
x > -5
Problem 8
Solve -4 < 2x ≤ 10.
Worked answer
-2 < x <= 5
-2 < x ≤ 5
Problem 9
Solve 1 ≤ x - 3 < 6.
Worked answer
4 <= x < 9
4 ≤ x < 9
Problem 10
"At most 12" translates to
- x ≤ 12
- x < 12
- x ≥ 12
- x > 12
Worked answer
x ≤ 12
It means no more than 12
So x ≤ 12
Problem 11
A lift holds at most 600 kg. Six people average 72 kg each. How much more weight can it take?
Worked answer
168
6 × 72 = 432 kg
600 - 432 = 168 kg
Problem 12
Solve x3 + 2 > 5.
Worked answer
x > 9
x3 > 3
x > 9
Problem 13
Entry costs $15 and each game is $3. With $45, how many games can you play?
Worked answer
10
3g ≤ 30
g ≤ 10
10 games
Problem 14
Solve 5x - 3 > 2x + 12.
Worked answer
x > 5
3x > 15
x > 5
Problem 15
A student has scores 78, 85 and 91. What must the fourth score be to average at least 85?
Worked answer
86
78 + 85 + 91 + s4 ≥ 85
254 + s ≥ 340
s ≥ 86
Problem 16
Solve -9 ≤ 4 - x < 1.
Worked answer
3 < x <= 13
-13 ≤ -x < -3
Multiply by -1: 13 ≥ x > 3
So 3 < x ≤ 13