Systems of Inequalities Test Answer Key with Scoring – Algebra 1

Question 1 Test points (2 points)

(a) 7 < 3(4) − 4 = 8 is true: solution.

(b) 2 < 3(2) − 4 = 2 is false (equal): not a solution.

Scoring: 1 point per part.

Question 2 Graph an inequality (2 points)

The boundary has y-intercept −1 and slope 2/3, so it passes through (0, −1) and (3, 1). It is solid. Test (0, 0): 0 ≤ −1 is false, so shade below the line (away from the origin).

Scoring: 1 point for a correct solid boundary, 1 point for the correct shading.

Question 3 Write the inequality (2 points)

The line passes through (0, 3) and (3, 0), so y = −x + 3. It is solid and the shading is above: y ≥ −x + 3.

y ≥ −x + 3

Scoring: 1 point for the boundary, 1 point for the symbol.

Question 4 Solve for y (3 points)

(a) −3y > −6x + 9. Divide by −3 and reverse: y < 2x − 3.

(b) Strict inequality: dashed.

(c) 0 < 2(0) − 3 = −3 is false, so (0, 0) is not in the solution set.

y < 2x − 3; dashed; no

Scoring: 1 point per part.

Question 5 Read a system (3 points)

(a) The solid line is y = x − 1 with shading above: y ≥ x − 1. The dashed line is y = −x + 5 with shading below: y < −x + 5.

(b) 2.5 ≥ 0.5 − 1 = −0.5 is true, and 2.5 < −0.5 + 5 = 4.5 is true.

y ≥ x − 1 and y < −x + 5

Scoring: (a) 1 point per inequality including the correct symbol; (b) 1 point.

Question 6 Choir trip (3 points)

(a) a + s ≤ 50 and 20a + 12s ≤ 800.

(b) (30, 20): 50 ≤ 50 is true, but 600 + 240 = 840 > 800: not feasible.

(c) (25, 25): 50 ≤ 50 and 500 + 300 = 800 ≤ 800: feasible.

Scoring: 1 point per part.

Question 7 Jewelry table (3 points)

(a) b + n ≥ 10, n ≤ 6, b ≥ 0, n ≥ 0.

(b) (4, 6): 10 ≥ 10 and 6 ≤ 6, feasible. (3, 6): 9 < 10, no. (5, 4): 9 < 10, no.

(c) For example (10, 0) or (8, 3).

Only (4, 6)

Scoring: 1 point per part; accept any valid extra point.

Question 8 No solution (2 points)

The lines are parallel (slope 4). The solution would need y above the upper line y = 4x + 9 and below the lower line y = 4x − 2, which is impossible. Algebraically it needs 4x + 9 < y < 4x − 2, so 9 < −2, which is false.

No solution

Scoring: 1 point for noting the parallel lines or the contradiction, 1 point for the clear explanation.