Pierre wants to make tables and chairs. He has a total of $220$ wooden boards and $760$ nails. $\newline$ $13T+8C \leq 220$ represents the number of tables $T$ and chairs $C$ he can make with $220$ wooden boards. $\newline$ $48T+37C \leq 760$ represents the number of tables and chairs he can make with $760$ nails. $\newline$ Does Pierre have enough boards and nails to make $9$ tables and $5$ chairs? $\newline$ Choose $1$ answer: $\newline$ (A) Pierre has enough boards and nails. $\newline$ (B) Pierre has enough boards but not enough nails. $\newline$ (C) Pierre has enough nails but not enough boards. $\newline$ (D) Pierre has neither enough boards nor enough nails.

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Algebra 1

Modeling with linear inequalities

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Q. Pierre wants to make tables and chairs. He has a total of

220

wooden boards and

760

nails.

\newline

13T+8C \leq 220

represents the number of tables

T

and chairs

C

he can make with

220

wooden boards.

\newline

48T+37C \leq 760

represents the number of tables and chairs he can make with

760

nails.

\newline

Does Pierre have enough boards and nails to make

9

tables and

5

chairs?

\newline

Choose

1

answer:

\newline

(A) Pierre has enough boards and nails.

\newline

(B) Pierre has enough boards but not enough nails.

\newline

\newline

(D) Pierre has neither enough boards nor enough nails.

Identify Constraints for Boards: Identify the constraints for the wooden boards. $\newline$ The inequality $13T + 8C \leq 220$ represents the maximum number of tables $(T)$ and chairs $(C)$ Pierre can make with the wooden boards he has.
Check Boards Availability: Substitute the values of $T$ and $C$ into the wooden boards inequality to check if Pierre has enough boards to make $9$ tables and $5$ chairs. $\newline$ For $T = 9$ and $C = 5$ , the inequality becomes $13(9) + 8(5) \leq 220$ . $\newline$ Calculate the left side of the inequality: $13 \times 9 + 8 \times 5 = 117 + 40 = 157$ .
Compare Boards Quantity: Compare the calculated value with the number of boards available. $\newline$ Since $157 \leq 220$ , Pierre has enough wooden boards to make $9$ tables and $5$ chairs.
Identify Constraints for Nails: Identify the constraints for the nails. The inequality $48T + 37C \leq 760$ represents the maximum number of tables ( $T$ ) and chairs ( $C$ ) Pierre can make with the nails he has.
Check Nails Availability: Substitute the values of $T$ and $C$ into the nails inequality to check if Pierre has enough nails to make $9$ tables and $5$ chairs. $\newline$ For $T = 9$ and $C = 5$ , the inequality becomes $48(9) + 37(5) \leq 760$ . $\newline$ Calculate the left side of the inequality: $48 \times 9 + 37 \times 5 = 432 + 185 = 617$ .
Compare Nails Quantity: Compare the calculated value with the number of nails available. $\newline$ Since $617 \leq 760$ , Pierre has enough nails to make $9$ tables and $5$ chairs.
Verify Resource Sufficiency: Determine if Pierre has enough resources to make $9$ tables and $5$ chairs. $\newline$ Since Pierre has enough wooden boards and enough nails, he can make $9$ tables and $5$ chairs.