Math · Algebra

Linear Inequalities in One or Two Variables on the Digital SAT

Inequalities are equations with a twist, and the twist is exactly where the SAT sets its traps. Solving 2x + 5 < 17 uses the same moves as solving 2x + 5 = 17, with one rule bolted on: multiply or divide both sides by a negative number and the inequality sign flips. Nearly every wrong answer on these questions is built around forgetting, or wrongly applying, that flip.

Because Algebra is the biggest Math domain on the Digital SAT, inequality questions are guaranteed to show up, and they come in two main flavors. The first is pure solving, sometimes with a negative coefficient waiting to catch you. The second is the constraint word problem: a budget, a weight limit, a minimum score, where you build the inequality yourself and then reason about which whole-number answers actually fit.

The word problems add one more wrinkle that pure algebra doesn't have: real-world rounding. If the math says you can afford 6.4 paperbacks, the answer is 6, because you can't buy 0.4 of a book. Knowing when to round down versus up is worth as many points as the algebra itself.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Solving one-variable linear inequalities, including ones that require flipping the inequality sign.
Solving compound (between) inequalities like a ≤ x ≤ b, including when every part is scaled by the same number.
Translating constraint situations like budgets, capacities, and minimum requirements into inequalities.
Finding the greatest or least whole-number value that satisfies a real-world inequality.
Identifying which values, or which points (x, y), satisfy an inequality or a system of two inequalities.
Writing a system of inequalities from a word problem with several constraints, turning each constraint into its own inequality.
Interpreting what an inequality's solution set means in context.
Matching a two-variable inequality to a described shaded region or set of solutions.
Counting the whole-number ordered pairs that satisfy a system of three or more inequalities in a feasible region.
Building the inequality for a boundary line from two points or a slope, then choosing the sign from an on, above, or below description.

Strategies that work

Flip only on negative multiply or divide

The inequality sign flips in exactly one situation: multiplying or dividing both sides by a negative number. Adding or subtracting anything, including negatives, never flips it. Say the rule to yourself at the moment you divide, because that is the single step where these questions are won or lost.

Translate the keywords literally

"At most" and "no more than" mean ≤, while "at least" and "no less than" mean ≥. Write the inequality directly from the sentence before doing any math. Students who jump straight to arithmetic often solve the right numbers with the wrong inequality direction.

Round the way reality demands

After solving a word problem, ask what the variable counts. If x ≤ 6.4 and x counts books you can afford, round down to 6, because 7 breaks the budget. If x ≥ 6.4 and x counts trips needed to haul everything, round up to 7. The direction of rounding comes from the story, not from the usual rounding rules.

Test a boundary number

Once you have an answer like x ≤ -5, plug the boundary value and one nearby value into the original inequality. If x = -5 works and x = -4 fails, your direction is right. This 15-second check catches flipped signs, the most common error on the entire skill.

Mistakes to avoid

Forgetting to flip the inequality sign when dividing both sides by a negative number.
Flipping the sign when it isn't needed, such as after subtracting a number from both sides.
Translating "at most" as < instead of ≤, which changes whether the boundary value counts.
Rounding up in a budget problem, choosing 7 items when only 6 fit under the limit.
Ignoring a fixed cost when setting up the inequality, for example dividing the whole budget by the per-item price.

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Linear Inequalities in One or Two Variables questions in a few clicks. Here is the move that applies, step by step.

Graph a System of Inequalities to Test a PointShaded regions overlap exactly where every inequality agrees.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

Consider the inequality 2x + 5 < 17.

Which of the following describes all solutions to the inequality?

Pick an answer to get instant coaching from Nova.

Question 2Medium

Consider the inequality -3x + 4 ≥ 19.

Which of the following describes all solutions to the inequality?

Pick an answer to get instant coaching from Nova.

Question 3Hard

Marisol has 60 dollars to spend at a book fair. She buys a tote bag for 12 dollars, and each paperback at the fair costs 7.50 dollars.

What is the greatest number of paperbacks Marisol can buy without going over her budget?

Pick an answer to get instant coaching from Nova.

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The rest of Algebra

Linear Inequalities in One or Two Variables is one of 4 skills in Algebra. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Algebra fits together.

Other Math skills to explore

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