Linear Inequalities

A linear inequality is like a linear equation, but with $<$, $>$, $\le$, or $\ge$ instead of $=$. Its solution is a range of values, not a single number. On the SAT you will solve one-variable inequalities (including compound ones), translate words like "at most" into symbols, and work with systems of inequalities, whose solutions are shaded regions in the $xy$-plane.

• Solve exactly as you would an equation, with one extra rule: multiplying or dividing by a negative number flips the inequality sign.
• A compound inequality such as $-5 < 1 - 3x \le 10$ is solved by doing the same step to all three parts at once.
• In context, the answer is often a whole number. "Greatest number of boxes" means round down; "least number of trips" means round up.
• A point is a solution to a system of inequalities only if it satisfies every inequality in the system.
• On a graph, $y > mx + b$ is the region above the line; $y < mx + b$ is below. A dashed line means the boundary is not included.

Example translation: a rental costs $40 plus $0.65 per mile, and the budget is at most $120. Then $40 + 0.65m \le 120$, so $m \le 123.07\ldots$, and the greatest whole number of miles is $123$.

Solve $5 - 3x < 17$.

1. Subtract $5$: $-3x < 12$.
2. Divide by $-3$ and flip the sign: $x > -4$.

Check with a value in the range, $x = 0$: $5 < 17$, true. Check one outside it, $x = -5$: $5 + 15 = 20 < 17$ is false. Correct.

The point $(a, 4)$ is a solution to the system $y \ge x - 3$ and $y < -2x + 16$. If $a$ is an integer, what is the greatest possible value of $a$?

1. Substitute $y = 4$ into the first inequality: $4 \ge a - 3$, so $a \le 7$.
2. Substitute into the second: $4 < -2a + 16$, so $2a < 12$ and $a < 6$.
3. Both must hold, so $a < 6$. The greatest integer is $a = 5$.

Check: $a = 5$ gives $4 \ge 2$ and $4 < 6$, both true. $a = 6$ gives $4 < 4$, which is false. So $5$ is the answer.

• Forgetting to flip. $-6x \le 18$ becomes $x \ge -3$, not $x \le -3$.
• Flipping when you shouldn't. Subtracting a negative number or dividing by a positive one never flips the sign.
• Rounding the wrong way. If $b \le 34.4$ boxes, the answer is $34$. If you need $t > 50$ minutes for something to happen, the least whole number is $51$.
• Using only one inequality in a system. A point that satisfies one condition but not the other is not a solution.
• Strict vs. inclusive. "At least" includes the boundary value; "more than" does not.