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A software company is hiring some programmers to add to its development team. A junior programmer's salary is $\$51,000$ and a senior programmer's salary is $\$78,000$ . To keep costs down, the total spending on these new positions must be under $\$960,000$ annually. $\newline$ Select the inequality in standard form that describes this situation. Use the given numbers and the following variables. $\newline$ $x =$ the number of junior programmers $\newline$ $y =$ the number of senior programmers $\newline$ Choices: $\newline$ (A) $51,000x + 78,000y \geq 960,000$ $\newline$ (B) $51,000x + 78,000y > 960,000$ $\newline$ (C) $51,000x + 78,000y < 960,000$ $\newline$ (D) $51,000x + 78,000y \leq 960,000$

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Write two-variable inequalities: word problems

Full solution

Q. A software company is hiring some programmers to add to its development team. A junior programmer's salary is

\$51,000

and a senior programmer's salary is

\$78,000

. To keep costs down, the total spending on these new positions must be under

\$960,000

annually.

\newline

Select the inequality in standard form that describes this situation. Use the given numbers and the following variables.

\newline

x =

the number of junior programmers

\newline

y =

the number of senior programmers

\newline

Choices:

\newline

(A)

51,000x + 78,000y \geq 960,000

\newline

(B)

51,000x + 78,000y > 960,000

\newline

(C)

51,000x + 78,000y < 960,000

\newline

(D)

51,000x + 78,000y \leq 960,000

Calculate Junior Programmers Cost: Determine the cost for hiring junior programmers. The salary for a junior programmer is $\$51,000$ per year, and the number of junior programmers is represented by $x$ . Therefore, the total cost for hiring $x$ junior programmers is $51,000x$ .
Calculate Senior Programmers Cost: Determine the cost for hiring senior programmers. The salary for a senior programmer is \$\(78\),\(000\) per year, and the number of senior programmers is represented by \(y\). Therefore, the total cost for hiring \(y\) senior programmers is \(78,000y\).
Combine Total Spending: Combine the costs for junior and senior programmers to represent the total spending on these new positions. The total spending is the sum of the cost for junior programmers and the cost for senior programmers, which is \(51,000x + 78,000y\).
Apply Spending Constraint: Apply the constraint that the total spending must be under \(\$960,000\) annually. This means that the sum of the salaries for junior and senior programmers must be less than \(\$960,000\). Therefore, the inequality that represents this situation is \(51,000x + 78,000y < 960,000\).

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