Inequalities flashcards

Only when you multiply or divide both sides by a negative number: −3x > 12 becomes x < −4. Adding, subtracting, or multiplying or dividing by a positive number keeps the direction.

No. You divide by positive 5, so x < −4. A negative number sitting on one side does not cause a flip; only a negative multiplier or divisor does.

Open circle for strict signs < and >: the boundary is not a solution. Closed, filled circle for ≤ and ≥: the boundary is included. Then shade the side that makes the inequality true.

Left endpoint, then right. A bracket means included, a parenthesis means excluded, and infinity always takes a parenthesis. $-1\le x<5$ is [−1, 5), and x > 2 is $(2,\infty )$.

'And' needs both conditions, so the solution is the overlap: x > 0 and x ≤ 4 gives 0 < x ≤ 4. 'Or' needs at least one, so the solution joins both sets, as in x < −2 or x > 3.

Less than: one between-statement, −b < x − a < b. Greater than: two pieces joined by or, x − a < −b or x − a > b. Example: |x − 3| ≤ 5 gives −2 ≤ x ≤ 8.

Round down for a maximum (at most, cannot exceed) and up for a minimum (at least). 9 + 8n ≤ 70 gives n ≤ 7.625, so at most 7; 5t ≥ 32 gives t ≥ 6.4, so at least 7.

Apply each step to all three parts: add 1 to get −6 ≤ 2x < 10, then divide by 2 to get −3 ≤ x < 5. Dividing all parts by a negative reverses both symbols.