Solving Inequalities — Challenge
Solve them exactly like equations — with one extra rule that catches almost everyone.
What these problems 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
What is the greatest integer x for which x+5<12?
- 5
- 6
- 7
- 8
- 17
Worked answer
6
Subtract 5 from both sides: x < 7.
The inequality is strict, so 7 itself is not allowed and the greatest integer below it is 6.
Problem 2
Which of these values of x satisfies -3x>12?
- -5
- -4
- 0
- 4
- 5
Worked answer
-5
Divide both sides by -3. Dividing an inequality by a negative number reverses it, giving x < -4.
Of the options only -5 is less than -4.
Check directly: -3(-5) = 15, and 15 > 12.
Problem 3
What is the greatest integer x for which 2x≤ 10?
- 2
- 4
- 5
- 6
- 20
Worked answer
5
Divide both sides by 2: x≤ 5.
Here the inequality allows equality, so 5 itself is permitted and is the greatest such integer.
Problem 4
What is the least integer x for which 4x-3≥ 9?
- 1
- 2
- 3
- 4
- 12
Worked answer
3
Add 3 to both sides: 4x≥ 12.
Divide by 4: x≥ 3. Equality is allowed, so the least such integer is 3.
Problem 5
What is the least integer x for which x-4>1?
- 3
- 4
- 5
- 6
- 9
Worked answer
6
Add 4 to both sides: x > 5.
The inequality is strict, so 5 does not count and the least integer above it is 6.
Problem 6
What is the least integer x for which 5-2x<1?
- 1
- 2
- 3
- 4
- 6
Worked answer
3
Subtract 5 from both sides: -2x < -4.
Divide by -2 and reverse the inequality: x > 2.
The least integer greater than 2 is 3. Check: 5-2(3) = -1, which is less than 1.
Problem 7
How many integers x satisfy -3≤ x<4?
- 6
- 7
- 8
- 9
- 10
Worked answer
7
The left end is included and the right end is not, so the integers are -3,-2,-1,0,1,2,3.
Counting them gives 7 integers. Including 4 by mistake would give 8.
Problem 8
What is the greatest integer x for which 3(x+2)≤ 18?
- 2
- 3
- 4
- 5
- 6
Worked answer
4
Divide both sides by 3: x+2≤ 6.
Subtract 2: x≤ 4, so the greatest such integer is 4.
Problem 9
A van may carry at most 900 kilograms. It already carries 340 kilograms of equipment. What is the greatest number of 35-kilogram boxes it can also carry?
- 14
- 15
- 16
- 17
- 18
Worked answer
16
If n is the number of boxes then 340+35n≤ 900.
Subtract 340: 35n≤ 560, so n≤ 16.
Sixteen boxes weigh 560 kilograms and bring the total to exactly 900, so 16 is allowed and 17 is not.
Problem 10
How many integers x satisfy -2<x+3≤ 5?
- 5
- 6
- 7
- 8
- 9
Worked answer
7
Subtract 3 from all three parts: -5 < x≤ 2.
The left end is excluded and the right end included, so the integers are -4,-3,-2,-1,0,1,2.
That is 7 integers.