➕ Algebra 1 · Inequalities
Solving Inequalities: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
The number 15 is a solution of x ≥ 15.
✓ True. 15 ≥ 15 is true, because ≥ means greater than or equal to.
The number 8 is a solution of x < 8.
✗ False. 8 < 8 is false: 8 is not less than itself. Numbers like 7 or 7.9 are solutions.
To solve −2x > 30, divide both sides by −2 and keep the sign, so x > −15.
✗ False. Dividing by −2 reverses the sign: x < −15. Test x = 0: −2(0) = 0 > 30 is false, but 0 > −15 is true, so x > −15 cannot be right.
You reverse the sign when you add a negative number to both sides.
✗ False. Only multiplying or dividing by a negative number reverses the sign. Adding −6 to both sides of x > 4 gives x − 6 > −2, with the same sign. (Check x = 5: −1 > −2 is true.)
If a < b, then −a > −b.
✓ True. Multiplying both sides by −1 reverses the sign. For example, 3 < 9 and −3 > −9.
The inequality 2x + 3 > 2x is true for every real number x.
✓ True. Subtract 2x from both sides: 3 > 0, which is true. The variable vanished and the leftover is true.
The inequality 5x − 1 < 5x − 6 has no solution.
✓ True. Subtract 5x from both sides: −1 < −6, which is false (−1 is to the right of −6).
The inequality x + 1 > x + 1 is true for every real number.
✗ False. Subtract x from both sides: 1 > 1, which is false, so there is no solution. With ≥ the leftover 1 ≥ 1 would be true.
On a number line, x ≤ 2 is drawn with an open circle at 2.
✗ False. The sign ≤ includes 2, so the circle is closed (filled).
x < 11 and 11 > x have the same solution set.
✓ True. Both say that x is less than 11.
A club has 100 dollars, and each shirt costs 15 dollars. If s is the number of shirts, then 15s ≤ 100 gives s ≤ 20/3, so the club can buy at most 6 shirts.
✓ True. 20/3 is between 6 and 7 (about 6.67). The solved inequality says s can be at most 20/3, and shirts come in whole numbers, so the greatest number that works is 6. Check: 6 shirts cost 90 dollars, and 7 shirts cost 105 dollars, which is too much.
Testing one number from the shaded part proves that a solution set is exactly right.
✗ False. One test can reveal an error, but it cannot prove the set is right. If the true answer is x > 4 and you wrote x > 2, the test number 6 passes both. Also test a number outside and the boundary.
Multiplying both sides of x/4 > −2 by 4 gives x < −8.
✗ False. The number 4 is positive, so the sign stays: x > −8. Test x = 0: 0 > −2 is true, and 0 > −8 is true.