A student showed their work for solving the inequality $-6(2 c-1)+30 \leq c-3$ for $c$.
| | $-6(2 c-1)+30 \leq c-3$ | |
| :--------- | :------------------------ | :---------- |
| Step One | $-12 c+6+30 \leq c-3$ | Distribute. |
| Step Two | $-12 c+36 \leq c-3$ | Combine like terms. |
| Step Three | $-13 c+36 \leq-3$ | Subtract c on both sides. |
| Step Four | $-13 c \leq-39$ | Subtract 36 on both sides. |
| Step Five | $c \leq 3$ | Divide by -13 on both sides. |
A student made an error in Step Five. What is the correct inequality for Step Five?
A. $c \leq 3$
B. $c \leq 507$
C. $c \geq 507$
D. $c \geq 3$
Answer (1)
Distribute and combine like terms to get − 12 c + 36 ≤ c − 3 .
Subtract c and 36 from both sides to obtain − 13 c ≤ − 39 .
Divide by -13, remembering to flip the inequality sign: c g e q − 13 − 39 .
Simplify to find the correct inequality: c g e q 3 .
Explanation
Analyze the problem Let's analyze the student's work step-by-step to identify the error. The initial inequality is − 6 ( 2 c − 1 ) + 30 ≤ c − 3 .
Step One Step One: Distribute the -6: − 12 c + 6 + 30 ≤ c − 3 . This step is correct.
Step Two Step Two: Combine like terms: − 12 c + 36 ≤ c − 3 . This step is also correct.
Step Three Step Three: Subtract c from both sides: − 13 c + 36 ≤ − 3 . This step is correct.
Step Four Step Four: Subtract 36 from both sides: − 13 c ≤ − 39 . This step is correct.
Step Five - Identifying the Error Step Five: Divide both sides by -13. Remember that when dividing by a negative number, we must flip the inequality sign. So, − 13 c ≤ − 39 becomes c g e q − 13 − 39 .
Correcting the Error Simplifying the fraction, we get c g e q 3 . The student made an error by not flipping the inequality sign, so the correct inequality for Step Five is c g e q 3 .
Final Answer Therefore, the correct inequality for Step Five is c g e q 3 .
Examples
Linear inequalities are used in everyday life to solve problems involving constraints. For example, suppose you have a budget of $100 to spend on groceries, and you want to buy apples that cost $2 per pound and bananas that cost $1 per pound. The inequality 2 a + b ≤ 100 represents the possible amounts of apples ( a ) and bananas ( b ) you can buy. Solving such inequalities helps you make informed decisions while staying within your budget.
