Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Solve 4x + 7 ≤ 2x − 5.
Show worked solution
2x + 7 ≤ −5, so 2x ≤ −12 and x ≤ −6. x ≥ −6 reverses the sign when nothing negative was divided; x ≤ −2 adds 2x instead of subtracting (6x ≤ −12); x ≤ 1 moves 7 across without changing its sign (2x ≤ 2).
2 Find the smallest integer that satisfies 2x + 9 > 4 − 3x, and the largest integer that satisfies 1 − 2x ≥ 3x − 14.
Show worked solution
5x > −5, so x > −1: −1 is left out, and the smallest integer is 0. 15 ≥ 5x, so x ≤ 3: the largest integer is 3.
3 What is the smallest integer that satisfies the inequality below?
Show worked solution
3x + 12 > 5, so 3x > −7 and x > −2⅓. The smallest integer above −2⅓ is −2. Test: 3(2) = 6 > 5 ✓, while x = −3 gives 3 > 5 ✗.
Level 2 · Core
4 Which inequality has the solution shown?
Show worked solution
6 − 2x < 10: −2x < 4, divide by −2 and reverse, x > −2 ✓. 6 − 2x > 10 gives x < −2, the wrong direction; 6 − 2x ≤ 10 gives x ≥ −2, a closed circle; 2x + 6 < 10 gives x < 2.
5 Solve −5 ≤ 3 − 2x < 7. Give the smallest and the largest integer in the solution.
Show worked solution
−8 ≤ −2x < 4. Divide by −2 and reverse: 4 ≥ x > −2, which is −2 < x ≤ 4. Smallest integer −1 (−2 is left out), largest 4. Test x = 4: 3 − 8 = −5 ≥ −5 ✓.
6 How many integers satisfy −9 < 4x + 3 ≤ 23?
Show worked solution
−12 < 4x ≤ 20, so −3 < x ≤ 5: the integers −2, −1, 0, 1, 2, 3, 4, 5, which is 8.
Level 3 · Challenge
7 Three statements:
1 If −x < 3, then x > −3.
2 x = 2 satisfies 3 − x ≥ 2x − 3.
3 2(x + 1) > 2x + 3 is true for every value of x.
Which statements are correct?
1 If −x < 3, then x > −3.
2 x = 2 satisfies 3 − x ≥ 2x − 3.
3 2(x + 1) > 2x + 3 is true for every value of x.
Which statements are correct?
Show worked solution
Multiplying −x < 3 by −1 reverses the sign: x > −3 (1 ✓). 3 − 2 = 1 and 4 − 3 = 1, and 1 ≥ 1 is true (2 ✓). 2x + 2 > 2x + 3 means 2 > 3, which is never true (3 ✗). 1 and 2.
8 x satisfies BOTH 5 − x < 1 AND 13 − 2x ≥ −7. Find the smallest and the largest integer x can be.
Show worked solution
−x < −4 gives x > 4. −2x ≥ −20 gives x ≤ 10. Together 4 < x ≤ 10: smallest 5, largest 10.
9 What is the smallest integer n for which the inequality below is true?
Show worked solution
× 15: 5(n − 4) < 3(2n + 1), so 5n − 20 < 6n + 3 and −23 < n. The end −23 is left out, so the smallest integer is −22. Test: −26/3 ≈ −8.67 < −43/5 = −8.6 ✓.