A software company is hiring some programmers to add to its development team. A junior programmer's salary is $55,000 and a senior programmer's salary is $85,000. To keep costs down, the total spending on these new positions must be under $510,000 annually.

Question

Write the inequality in standard form that describes this situation. Use the given numbers and the following variables.
x = the number of junior programmers
y = the number of senior programmers

Answer 1

To represent the situation mathematically, we can establish the following inequality based on the salary information provided.

Let:

• \( x \) = the number of junior programmers
• \( y \) = the number of senior programmers

The salary for a junior programmer is $55,000, so the total salary for junior programmers can be expressed as \( 55000x \).

The salary for a senior programmer is $85,000, so the total salary for senior programmers can be expressed as \( 85000y \).

According to the given requirement, the total spending on these new positions must be under $510,000 annually. This can be written as the inequality:

\[ 55000x + 85000y < 510000 \]

To convert this inequality into standard form (Ax + By < C), we can rearrange it:

\[ 55000x + 85000y < 510000 \]

This expression is already in a form that can be considered standard, but we can also multiply the entire inequality by a common factor to simplify, if needed. However, since all coefficients are already in a manageable form, the standard form of the inequality remains:

\[ 55000x + 85000y < 510000 \]

You could also express this with integer coefficients by dividing through by 5000 for simplicity:

\[ 11x + 17y < 102 \]

But the original inequality is perfectly valid as well. Therefore, one of the standard forms of the inequality that describes the situation is:

\[ 55000x + 85000y < 510000 \]