Solving Inequalities — Drills
Solving Inequalities has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Two-step
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 8x - 5 ≥ 27. Enter it like x>=4.
Worked answer
x>=4
8x ≥ 32
x ≥ 4
Problem 2
Solve 2x + 9 ≥ 11. Enter it like x>=1.
Worked answer
x>=1
2x ≥ 2
x ≥ 1
Flip
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve -8x - 5 ≤ -45. Enter it like x>=5.
Worked answer
x>=5
-8x ≤ -40
Divide by -8 and flip: x ≥ 5
Problem 2
Solve -2x + 9 > 5. Enter it like x<2.
Worked answer
x<2
-2x > -4
Divide by -2 and flip: x < 2
One-step
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x - 10 > 6. Enter it like x>16.
Worked answer
x>16
x > 16
Problem 2
Solve x - 8 ≥ -17. Enter it like x>=-9.
Worked answer
x>=-9
x ≥ -9
Notation
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
On a number line, which dot do you use for x ≤ 3?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
A closed (filled) dot
That symbol includes the endpoint, so 3 is a solution and the dot is closed.
Problem 2
On a number line, which dot do you use for x ≥ 0?
- A closed (filled) dot
- An open (hollow) dot
Worked answer
A closed (filled) dot
That symbol includes the endpoint, so 0 is a solution and the dot is closed.
Parens
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 2(x + 1) ≤ 10. Enter it like x<=4.
Worked answer
x<=4
x + 1 ≤ 5
x ≤ 4
Problem 2
Solve 3(x - 7) > -15. Enter it like x>2.
Worked answer
x>2
x - 7 > -5
x > 2
Word problems
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Rosa has $120. After buying a $28 jacket, T-shirts cost $9 each. At most how many T-shirts?
Worked answer
10
9t ≤ 92
t ≤ 10.22
Part of a shirt is no use, so 10.
Problem 2
Devon has $60. After buying a $25 jacket, T-shirts cost $15 each. At most how many T-shirts?
Worked answer
2
15t ≤ 35
t ≤ 2.33
Part of a shirt is no use, so 2.
Bothsides
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 5x + 10 ≥ 3x + 18. Enter it like x>=4.
Worked answer
x>=4
2x + 10 ≥ 18
2x ≥ 8
x ≥ 4
Problem 2
Solve 6x + 5 < 3x - 13. Enter it like x<-6.
Worked answer
x<-6
3x + 5 < -13
3x < -18
x < -6
Compound
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many integers satisfy -7 ≤ 3x - 4 < 17?
Worked answer
8
Add 4 everywhere: -3 ≤ 3x < 21
Divide by 3: -1 ≤ x < 7
Integers from -1 to 6
That is 8 values — note 7 is excluded.
Problem 2
How many integers satisfy -14 ≤ 4x - 2 < 6?
Worked answer
5
Add 2 everywhere: -12 ≤ 4x < 8
Divide by 4: -3 ≤ x < 2
Integers from -3 to 1
That is 5 values — note 2 is excluded.