Linear and Compound Inequalities: Algebra 1 Math Test Answer Key – PDF Download

Question 1: Solve and graph (2 points)

Subtract \(3x\): \(2x-12>6\). Add 12: \(2x>18\). Divide: \(x>9\).

Scoring: 1 point for the correct inequality, 1 point for the graph with an open circle at 9 and a ray to the right.

Question 2: Distribute with a negative (2 points)

Distribute: \(-3x+12\le2x+7\). Subtract \(2x\): \(-5x+12\le7\). Subtract 12: \(-5x\le-5\). Divide by \(-5\) and reverse: \(x\ge1\). Check: \(x=1\) gives \(9\le9\).

Scoring: 1 point for distributing and isolating the variable term correctly, 1 point for reversing the sign to get the correct answer.

Question 3: Negative fraction (2 points)

Subtract 2: \(-\frac{3}{4}x>6\). Multiply by \(-\frac{4}{3}\) and reverse: \(x<-8\). Check: \(x=-12\) gives \(9+2=11>8\).

Scoring: 1 point for isolating the term, 1 point for the correct answer with the sign reversed.

Question 4: Read a compound graph (2 points)

\(-2\le x< 5\), interval notation [-2, 5).

Scoring: 1 point for the inequality with the correct symbols, 1 point for interval notation with a bracket and a parenthesis.

Question 5: Three-part inequality (3 points)

Add 1 to all parts: \(-6\le3x< 12\). Divide by 3: \(-2\le x< 4\).

Scoring: 2 points for the solution (1 point if only one bound is correct), 1 point for the graph with a closed circle at -2 and an open circle at 4.

Question 6: Either-or inequality (3 points)

First: \(3x\le-9\), so \(x\le-3\). Second: \(2x>8\), so \(x>4\). Solution: \(x\le-3\text{ or }x>4\).

Scoring: 2 points for solving both parts (1 point each), 1 point for the correct graph.

Question 7: Test a value (2 points)

1. Substitute: \(6-2(-2)=10\), and \(10\ge10\) is true, so yes.
2. Subtract 6: \(-2x\ge4\). Divide by \(-2\) and reverse: \(x\le-2\).

Scoring: 1 point for the substitution and conclusion, 1 point for the solution with the sign reversed.

Question 8: County fair budget (3 points)

1. \(12+2.50t\le40\)
2. Subtract 12: \(2.50t\le28\). Divide: \(t\le11.2\).
3. Tickets are whole numbers, so 11 tickets. Check: \(12+2.50(11)=39.50\).

Scoring: 1 point for the inequality, 1 point for solving, 1 point for rounding down to 11 with a reason.

Question 9: Special case (1 point)

Distribute: \(2x+6< 2x+4\). Subtract \(2x\): \(6< 4\), which is false. No solution.

Scoring: 1 point for concluding that there is no solution with the false statement shown.

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