SAT Math · Algebra · Skill 5 of 5

Linear inequalities

Not one answer but a whole range of them. Learn the one rule that trips everyone up, turn words into symbols, and shade the right side of the line every time.

8 chapters plus a challengeAbout 35 minutesFree, no sign-up
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Inequalities on a number line

Explore · 4 min

An inequality describes a range of values. On a number line, an open circle means the boundary is not included (< or >), and a filled circle means it is included (≤ or ≥). The shading shows every value that works.

Solving, and the one rule that flips

Practise · 6 min

Solve inequalities exactly like equations, with one extra rule: when you multiply or divide both sides by a negative number, flip the sign. Press the button to see why.

Worked example: −3x + 4 ≥ 13
  1. Subtract 4 from both sides: −3x ≥ 9 (no flip for subtracting)
  2. Divide both sides by −3 and flip the sign: x ≤ −3
  3. Check with x = −4: −3(−4) + 4 = 16, and 16 ≥ 13. It works.

Words to symbols

Game · 3 min

Word problems hide the symbol in a phrase. The key question: is the boundary value itself allowed? If yes, the symbol has a line under it.

Compound inequalities

Learn · 4 min

A compound inequality like −1 ≤ 2x + 3 < 9 squeezes x between two values. Solve it by doing the same operation to all three parts at once.

−1 ≤ 2x + 3 < 9
  1. Subtract 3 from all three parts: −4 ≤ 2x < 6
  2. Divide all three parts by 2: −2 ≤ x < 3
  3. The integers that work are −2, −1, 0, 1 and 2. The −2 is included (≤) but 3 is not (<).

Modelling constraints

Scenarios · 6 min

Budgets, weight limits and target scores are all inequalities. The answer is usually a maximum or minimum whole number, so read carefully: you cannot buy 5.6 tickets.

Two variables: shade a half-plane

Explore · 5 min

An inequality in x and y, like y > 2x − 1, is true for a whole region. Draw the boundary line (dashed for < and >, solid for ≤ and ≥), then shade above for greater than and below for less than. Pick a symbol and drag the lens.

Test a point

Practise · 3 min

The fastest way to answer “which point satisfies” questions: substitute each option. For a system of inequalities the point must satisfy every inequality, and it lives where the shaded regions overlap.

Spot the trap

Detective · 3 min

Each solution has one faulty step. Two of them are the most common inequality mistakes on the SAT.

Final challenge

8 questions · 8 min

Key ideas to carry in
Solve like an equationSame moves, same order.
The flip ruleMultiply or divide by a negative: flip the sign.
Included or not≤ and ≥ include the boundary; < and > do not.
CompoundDo the same thing to all three parts.
GraphsGreater than: shade above. Strict: dashed line.
Word problemsThe answer is often a whole-number maximum or minimum.

Desmos works here tooType an inequality such as y > 2x − 1 and Desmos shades the region for you. Type two and the overlap shows the solutions to the system.

Challenge