Solving Inequalities: notes and common mistakes

Key ideas

• An inequality compares two expressions with <, >, ≤ or ≥. Its solution set is usually infinite, so we show it on a number line.
• Open circle: the boundary is not included (<, >). Closed circle: the boundary is included (≤, ≥).
• Adding or subtracting the same number on both sides, or multiplying or dividing both sides by a positive number, keeps the sign. Multiplying or dividing both sides by a negative number reverses it.
• If the variable vanishes, a true leftover (like 1 > 0) means every real number is a solution, and a false leftover (like 1 < 0) means no solution.
• “At least” is ≥, “at most” is ≤, “more than” is > and “fewer than” is <. Define the variable first, and say what the answer means in the story.

Steps

1. Define the variable (in word problems) and write the inequality.
2. Simplify each side: distribute and combine like terms.
3. Gather the variable terms on one side and the plain numbers on the other. The sign stays.
4. Multiply or divide both sides to free the variable. If the number is negative, reverse the sign.
5. Graph it: an open or closed circle at the boundary, and shade the true side.
6. Check with a number from the shaded part, a number from outside, and the boundary.

Formulas

• Add, or multiply by a positive: sign stays: If a < b, then a + c < b + c for any number c. If also c > 0, then ac < bc.
• Negative multiplier: sign reverses: If a < b and c < 0, then ac > bc and a/c > b/c
• Words to symbols: at least: ≥. at most: ≤. more than: >. fewer than: <

Watch out

• Reverse the sign only when you multiply or divide both sides by a negative number. A negative number somewhere else in the problem does not reverse it.
• Adding or subtracting never reverses the sign: x − 3 > −8 becomes x > −5.
• < and > get an open circle. ≤ and ≥ get a closed circle.
• Put x on the left before you shade: 3 < x means x > 3, so shade to the right.
• When the variable vanishes, read the leftover statement. 1 > 1 is false (no solution), but 1 ≥ 1 is true (every real number).
• In a word problem, interpret the answer. When only whole numbers make sense, read the solved inequality: for m ≤ 5.4 pick the greatest whole number that works (5), and for m ≥ 5.4 pick the least (6). Then test it in the original story.

Why it works

• Why does the sign reverse? If a < b, then b − a is positive. A negative number c times a positive number is negative, so c(b − a) < 0. That means cb − ca < 0, so cb < ca, which is the same as ca > cb.
• Why can you add the same number to both sides? Adding slides both numbers the same distance along the number line, so their order does not change.
• Why does a true leftover mean every real number works? When the variable terms cancel, the comparison no longer depends on x. It is the same true (or false) statement for every x.