Linear Inequalities in One or Two Variables Flashcards

Which of the following is equivalent to the inequality −x > 4 − 3x?

Answer: x > 2

Add 3x to both sides: 2x > 4 → x > 2.

A tank starts with 400 liters and is drained at 15 liters per minute. How many complete minutes m can it drain before falling below 100 liters?

Answer: 19

400 − 15m ≥ 100 → 15m ≤ 300 → m ≤ 20; the tank is at or above 100 L for 20 complete minutes, meaning it is still at exactly 100 after minute 20 — for 19 complete minutes it is strictly above 100.

If f(x) = 5 − 2x and f(x) > 1, what is the solution set?

Answer: x < 2

5 − 2x > 1 → −2x > −4 → x < 2.

The shaded region on a graph is above the line y = 2x − 3 and below y = 5. Which system describes it?

Answer: y > 2x − 3 and y < 5

Above (strictly) means y > 2x − 3; below (strictly) means y < 5.

Solve: |2x − 3| ≤ 7 (treat as a linear inequality by splitting into compound form)

Answer: −2 ≤ x ≤ 5

−7 ≤ 2x − 3 ≤ 7 → −4 ≤ 2x ≤ 10 → −2 ≤ x ≤ 5.

For which value of c does the inequality 3x + c ≤ 15 have x = 4 as its maximum integer solution?

Answer: c = 3

x = 4 is max integer, so 4 satisfies but 5 does not: 3(4)+c ≤ 15 → c ≤ 3, and 3(5)+c > 15 → c > 0; c = 3 satisfies both.

Owen needs at least $500 for a trip. He has saved $80 and will earn $15/hr this weekend. How many hours h must he work?

Answer: h ≥ 28

80 + 15h ≥ 500 → 15h ≥ 420 → h ≥ 28.

Which inequality is represented by the graph showing a dashed line through (0, 4) and (2, 0), with shading below?

Answer: y < −2x + 4

The slope is (0−4)/(2−0) = −2, y-intercept = 4; dashed line means strict inequality; shading below means y < −2x + 4.