Solve and interpret linear inequalities in one or two variables, including constraint word problems.
You solve inequalities, translate constraints into inequalities, and read solution regions. One-variable questions ask for the greatest or least integer that satisfies a condition. Two-variable questions ask which inequality or system of inequalities models a situation with limits, such as a budget cap or a minimum requirement, or which ordered pair falls in a shaded region. Words like at least, at most, no more than, and minimum control the direction of the sign and are the whole point of many of these questions.
Linear inequalities usually account for about 2 to 3 of the 44 math questions, within the roughly 35 percent Algebra share.
Written for this skill, with the answer explained. 109 more are in the app.
In the xy-plane, the shaded region shows all solutions to the inequality y ≤ -2x + 6. What is the greatest possible value of y when x = 1?
Solve -(x - 5) > 2.
Why: Distribute the minus sign: -x + 5 > 2. Subtract 5 from both sides: -x > -3. Multiply by -1, which reverses the inequality: x < 3. Answering x > 3 keeps the original direction after multiplying by a negative. Answering x < 7 and x > 7 come from adding 5 and 2 instead of subtracting, and testing x = 7 gives -(2) = -2, which is not greater than 2.
The solution of 2x + 3 > 5 and x - 1 < 4 is
Why: Solve each part. From 2x + 3 > 5, 2x > 2 so x > 1. From x - 1 < 4, x < 5. Both must hold at once, so the solution is 1 < x < 5. Answering x > 1 uses only the first inequality and allows values like 10 that break the second. Answering x < 1 reverses the first inequality, and x > 5 lies entirely outside the second.
A coach needs at least 150 protein bars and already has 42 of them. Bars come only in boxes of 24. What is the least number of boxes the coach must order?
Why: The coach needs 150 - 42 = 108 more bars, and 108/24 = 4.5 boxes. Boxes come whole and 4 boxes give only 96 more bars, so 5 boxes are needed. Choice A rounds down and falls short. Answering 4 means you rounded 4.5 down, which leaves fewer than 150 bars. Answering 6 means you ignored the 42 bars on hand and rounded 6.25 down. Answering 7 means you ignored the 42 bars on hand and rounded 6.25 up.
Only when you multiply or divide both sides by a negative number. Adding or subtracting a negative number does not flip anything. If you never divide by a negative, the sign stays the way it started.
At most and no more than mean less than or equal to. At least and no fewer than mean greater than or equal to. Fewer than and more than are strict, with no equal-to part.
Pick a test point that is clearly inside one candidate region, often the origin if it is not on a boundary line, and plug it into every inequality. The correct system is the one all of whose inequalities the point satisfies when it should, and fails when it should not.
Questions on linear inequalities alone, with explanations on every one, and the app tracks which ones you keep missing.
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