Inequalities
Inequalities
- An inequality uses $<$, $>$, $\leq$ or $\geq$. Solve like an equation — but reverse the sign if you multiply or divide by a negative.
- Solve $-3 \leq 3x - 2 < 7$: add 2 → $-1 \leq 3x < 9$ → divide by 3 → $-\tfrac13 \leq x < 3$.
Practice
Solve −3 ≤ 3x − 2 < 7. The solution is −1/3 ≤ x < ? What is the upper bound?
Add 2: −1 ≤ 3x < 9; divide by 3: −1/3 ≤ x < 3.
Practice
You must reverse the inequality sign when you:
Multiplying or dividing by a negative flips < to > (and vice versa).
On a number line
- Open circle for $<$ or $>$ (not included); closed circle for $\leq$ or $\geq$ (included).
- (Extended) An inequality in two letters shades a region (broken boundary for strict, solid otherwise).
Practice
On a number line, the symbol ≤ is drawn with a:
≤ and ≥ include the end value, shown by a closed circle; < and > use an open circle.
You've got it
Key idea
- solve like an equation, but flip the sign when multiplying/dividing by a negative
- $-3 \leq 3x - 2 < 7 \Rightarrow -\tfrac13 \leq x < 3$
- number line: open circle excludes, closed circle includes the end