A shipping container initially holds 450 packages. A crew unloads the container at a rate … — SAT

A shipping container initially holds 450 packages. A crew unloads the container at a rate of 15 packages per hour, while an automated sorting machine unloads packages at a rate of $x$ packages per hour. The number of packages remaining in the container after 8 hours must be at most 130. The inequality $450 - 8(15 + x) \leq 130$ models this scenario. What is the minimum possible value of $x$ ?

Answer: 25

Answer

The minimum possible value of

x

is 25.

To find the minimum possible value of

x

, solve the inequality

450 - 8(15 + x) \leq 130

. First, distribute the

-8

to obtain

450 - 120 - 8x \leq 130

. Simplify the constant terms on the left side to get

330 - 8x \leq 130

. Subtract 330 from both sides, yielding

-8x \leq -200

. Finally, divide both sides by

-8

and reverse the inequality sign because of the division by a negative number, which results in

x \geq 25

. The minimum possible value of

x

is 25.

Step-by-Step Solution

Distribute the factor of

-8

to both terms inside the parentheses.

450 - 120 - 8x \leq 130

To remove the parentheses and prepare to combine like terms.

Subtract 120 from 450 to simplify the constants on the left side.

330 - 8x \leq 130

To simplify the left side of the inequality before isolating the variable.

Subtract 330 from both sides of the inequality.

-8x \leq -200

To isolate the variable term on the left-hand side.

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq 25

Dividing by a negative value reverses the inequality sign. Since

x

must be greater than or equal to 25, the minimum possible value is 25.

Key Concept

Solving multi-step linear inequalities in one variable, including distributing coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

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