Educerie · SAT · Math
Algebra · ALG.5 Linear inequalities in one or two variables
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Solve a linear inequality | …which of the following is a solution to 4 − 2x ≥ 10? |
| Flip the sign correctly | Multiply or divide by a negative and the inequality reverses |
| Translate a constraint into an inequality | at least, no more than, at most, fewer than |
| Build a system of constraints | Two conditions on the same pair of variables |
| Read a shaded region | Which side of the line, and whether the line itself counts |
1The idea in one paragraph
Solve an inequality exactly as you would the equation — the same moves, in the same order — with one rule added: multiplying or dividing both sides by a negative number reverses the sign. The rest of this unit is translation. Real constraints arrive as English, and most of the marks here are lost turning at least 40 into ≥ 40 rather than doing any algebra at all.
A quick check that costs three seconds: take any number from your answer and put it back in the original inequality. If it does not satisfy it, you flipped when you should not have, or did not when you should.
2The one rule
Why it happens: −2 is less than 3, but multiplying both by −1 gives 2 and −3, and now the first is greater. The order reverses.
Adding and subtracting never flip anything. Multiplying or dividing by a positive never flips anything. Only the negative does.
3The words, and what they mean
| English | Symbol |
|---|---|
| at least, no fewer than, a minimum of | ≥ |
| at most, no more than, a maximum of, up to | ≤ |
| more than, exceeds, over | > |
| fewer than, under, below | < |
At least and at most include the boundary; more than and fewer than do not. On the test that distinction usually decides between two answer choices that are otherwise identical, and it is worth reading the sentence twice for.
4Building a constraint from a context
A van can carry no more than 1,200 kg. Each crate weighs 35 kg and the driver weighs 80 kg.
An inequality question about a countable thing nearly always wants the whole number at the edge.
That final step is where the points go. The algebra gives 32; a context about people or crates cannot take 32.5.
5Systems of inequalities and their regions
Two constraints at once describe an overlapping region. On the test you are usually asked which point satisfies both, which is answered by substitution: try the point in each inequality and keep the one that survives both.
When a graph is given: a solid line means the boundary is included (≤ or ≥), a dashed line means it is not (< or >). Shading above the line goes with y >, shading below with y <.
6The calculator draws the region
Typing an inequality into the built-in graphing calculator shades the region directly. For a system, type both and the overlap is visible. On a question asking which ordered pair satisfies a system, this is faster than four substitutions.
Where points are lost
- Forgetting to flip when dividing by a negative — the single most common error in this unit.
- Flipping when adding a negative number, which changes nothing.
- Confusing at least with more than, and so including or excluding the boundary wrongly.
- Leaving a fractional answer in a context that counts objects.
- Shading the wrong side because the inequality was not rearranged into y form first.
- Solid versus dashed on a graph.
Work it right
- Translate the English into a symbol, checking whether the boundary is included.
- Solve exactly as you would an equation.
- Flip the sign if and only if you multiplied or divided by a negative.
- Test one number from your answer in the original.
- If the context counts things, round to the whole number on the correct side.
Try it
Q1. What is the solution to 7 − 3x > 22?
A) x > −5 B) x < −5 C) x > 5 D) x < 5
Q2. A lift can carry a maximum of 900 kg. Six people weighing 480 kg in total are already inside, and boxes weighing 35 kg each are to be loaded. What is the greatest number of boxes that can be loaded? (Type your answer.)
Q3. Which ordered pair satisfies both y > 2x − 3 and y ≤ −x + 6?
A) (4, 5) B) (1, 2) C) (3, 4) D) (0, −4)
Q4. A student needs an average of at least 80 across four tests. Their first three scores are 74, 88 and 79. What is the lowest score they can get on the fourth test and still reach the average?
A) 77 B) 79 C) 80 D) 82
Q5. The inequality −4x + 9 ≤ 1 is equivalent to which of the following?
A) x ≤ 2 B) x ≥ 2 C) x ≤ −2 D) x ≥ −2
In one breath
Solve an inequality exactly like the equation and flip the sign only when you multiply or divide by a negative — then test one number from your answer in the original to be sure. Most of the points here are lost in translation rather than algebra: at least and at most include the boundary while more than and fewer than do not, and a context that counts crates or people needs the whole number at the edge of the range.
Answers
Q1. B — x < −5. divide by −3 and flip
A forgets the flip. C and D lose the negative when dividing.
Q2. 12. build the inequality, then round down
Twelve boxes bring the load to exactly 900 kg, which is allowed because the limit is a maximum. Answering 13 ignores the constraint; answering 11 forgets that the boundary is included.
Q3. B — (1, 2). test in both 2 > 2(1) − 3 = −1 ✓ and 2 ≤ −1 + 6 = 5 ✓. A: 5 > 5 is false. C: 4 ≤ 3 is false. D: −4 > −3 is false.
Q4. B — 79. turn the average into a total
A is one short; C and D are higher than necessary, and the question asked for the lowest.
Q5. B — x ≥ 2. subtract 9, then divide by −4 and flip
A forgets the flip. C and D mishandle the sign of −8.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Algebra, skill/knowledge testing point "Linear inequalities in one or two variables"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
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