Gwen is booking a band and a DJ for a dance, and wants to keep the cost of the entertainment under $750. A band can play for part of the time at an hourly rate of $170, and a DJ can play for the remaining time at an hourly rate of $70.

Question

Write the inequality in standard form that describes this situation. Use the given numbers and the following variables.
x = the number of hours the band will play
y = the number of hours the DJ will play

Answer 1

To describe the situation mathematically, we can set up the following inequality based on the costs associated with hiring the band and the DJ.

The total cost of hiring the band for \( x \) hours at a rate of $170 per hour is given by \( 170x \). Similarly, the total cost of hiring the DJ for \( y \) hours at a rate of $70 per hour is represented by \( 70y \).

Gwen wants to keep the total cost under $750. Therefore, we can establish the following inequality:

\[ 170x + 70y < 750 \]

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

1. Move all terms to one side of the inequality:

\[ 170x + 70y - 750 < 0 \]

2. To write it in standard form, we can multiply through by -1 (which will flip the inequality):

\[ -170x - 70y + 750 > 0 \]

For standard form, we typically want the coefficients of \( x \) and \( y \) to be positive. Thus, the preferred standard form of the inequality would be:

\[ 170x + 70y < 750 \]

Thus, the final standard form of the inequality is:

\[ 170x + 70y < 750 \]