The four symbols, and what the circle looks like.
| Symbol | Means | Circle |
|---|---|---|
| < | less than | open — the value is excluded |
| > | greater than | open |
| ≤ | less than or equal to | closed — the value is included |
| ≥ | greater than or equal to | closed |
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- Grade 7-12 Math Review
- Linear Inequalities in One Variable
Linear Inequalities in One Variable
Solve it exactly like an equation — until you divide by a negative.
Solve for x
−4x + 9 ≥ 25
Subtract 9: −4x ≥ 16. Divide by −4 and flip the sign: x ≤ −4. B forgets the flip, which is the whole point of the lesson. Test x = −5: −4(−5) + 9 = 29 ≥ 25 ✓ and −5 ≤ −4 ✓ — so the answer must include numbers below −4, which only C does.
Adding and subtracting change nothing.
x + 7 < 12
the inequality
x < 5
− 7 both sides, sign unchanged
Moving everything 7 places along the number line does not change which one is further left. The order survives.
Multiplying or dividing by a negative reverses it.
This is the only rule that makes inequalities different from equations.
−4x + 9 ≥ 25
the opening problem
−4x ≥ 16
− 9, sign unchanged
x ≤ −4
÷ (−4), sign flipped
Only × and ÷, never + and −
Subtracting a negative number does not flip the sign. It is multiplying or dividing by a negative that reverses the order.
Avoid the flip entirely, if you prefer.
−4x + 9 ≥ 25
the same problem
9 ≥ 25 + 4x
+ 4x both sides
−16 ≥ 4x
− 25
−4 ≥ x, so x ≤ −4
÷ 4, positive — no flip
Keeping the variable term positive means never dividing by a negative. Same answer, one fewer thing to remember.
Test a number. It settles everything.
x = −5
−4(−5) + 9 = 29 ≥ 25
works, and −5 ≤ −4 ✓
x = −4
16 + 9 = 25 ≥ 25
the endpoint counts ✓
x = 0
9 ≥ 25 is false
so x ≥ −4 must be wrong ✗
Pick a number clearly inside your answer and one clearly outside it. If both behave as they should, the direction of the sign is right.
Three-part inequalities, and word problems that need whole numbers.
Whatever you do to the middle, do to all three parts — and flip both signs for a negative.
Solve 1 ≤ 5 − 2x < 9.
1 ≤ 5 − 2x < 9
three parts
−4 ≤ −2x < 4
− 5 from all three
2 ≥ x > −2
÷ (−2): both signs flip
−2 < x ≤ 2
rewrite smallest first
Test x = 0, inside: 5 − 0 = 5, and 1 ≤ 5 < 9 ✓. On a number line this is an open circle at −2, a closed one at 2, and shading between.
A store owner buys a box of noodles for ₱450 and sells each pack for ₱16. How many packs must she sell to cover the cost?
16p ≥ 450
"cover the cost" means at least ₱450
p ≥ 28.125
÷ 16
p = 29 packs
the next whole number
Rounding down when you need "at least"
28 packs bring in 16 × 28 = ₱448, still ₱2 short. When the answer must reach a target, round up to the next whole number: 16 × 29 = ₱464 ✓.
Variations: fractions, always true, and rounding down.
The same rules; three answers that look different.
2x − 13 ≤ x2 + 1
2(2x − 1) ≤ 3x + 6
× 6, positive: no flip
4x − 2 ≤ 3x + 6
expand
x ≤ 8
− 3x, + 2
Check the endpoint: 16 − 13 = 5 and 82 + 1 = 5, equal ✓. Clearing fractions by a positive number never flips the sign.
3(x + 2) > 3x + 1
3x + 6 > 3x + 1
expand
6 > 1
the x terms cancel
6 > 1 is always true, so every real number is a solution. Had it ended in something false, like 6 > 9, there would be no solution at all.
Lea has ₱500. A notebook costs ₱45. How many can she buy?
45n ≤ 500
she can spend at most ₱500
n ≤ 11.1…
÷ 45
n = 11 notebooks
the largest whole number
Here we round down: 12 notebooks cost ₱540, more than she has. "At most" rounds down, "at least" rounds up.
On the exam
Periodical tests ask you to solve, to match the answer to a number-line graph, or to find the least or greatest whole number in a word problem. The fastest route is to solve like an equation, flip only when you multiply or divide by a negative, and read "at least" as ≥ and "at most" as ≤. For a 10-second check, put one number from inside your answer into the original.
Now you try
Practice problem
Solve for x
7 − 3x > 19
Subtract 7: −3x > 12. Divide by −3 and flip: x < −4. Test x = −5: 7 + 15 = 22 > 19 ✓ Test x = 0: 7 > 19 is false, so the answer cannot include 0 — which rules out A immediately. A keeps the original direction after dividing by a negative.
Hint: the x term is negative, so expect a flip. Then test x = 0 — if it fails, your answer must not include it.