Chapter 9.5, Problem 71SR

To determine

Write an inequality that represents the situation and graph the solution.

Answer to Problem 71SR

  $12x+3y\le 60$

  

Explanation of Solution

Given information:

A large pizza cost $12 and pitcher of a soft drink cost $3. she does not want to spend more than $60.

Concept used:

Transformation of functions: For any positive integer a , we have

• $y=f\left(x+a\right)$ is the graph of $y=f\left(x\right)$ shifted a units left.
• $y=f\left(x-a\right)$ is the graph of $y=f\left(x\right)$ shifted a units right.
• $y=f\left(x\right)+a$ is the graph of $y=f\left(x\right)$ shifted a units up.
• $y=f\left(x\right)-a$ is the graph of $y=f\left(x\right)$ shifted a units down.
• $y=af\left(x\right)$ is the graph of $y=f\left(x\right)$ stretches a units in y direction if $a>1$ and compresses a units in y direction if $0<a<1$
• $y=f\left(ax\right)$ is the graph of $y=f\left(x\right)$ stretches a units in x direction if $0<a<1$ and compresses a units in x direction if $a>1$ .
• $y=-f\left(x\right)$ is the graph of $y=f\left(x\right)$ reflected about x axis.
• $y=f\left(-x\right)$ is the graph of $y=f\left(x\right)$ reflected about y axis.

Calculation:

Let x = the number of pizzas she can buy

Let y = the number of pitchers of soft drink she can buy

So, equation is:

  $12x+3y\le 60$

Solving the equation to get the value of y , first subtract 12x from both sides, and then divide both sides by 3 as shown below:

  $\begin{array}{l}12x+3y\le 60\\ 12x-12x+3y\le 60-12x\\ 3y\le 60-12x\\ \frac{3y}{3}\le \frac{60}{3}-\frac{12x}{3}\\ y\le 20-4x\end{array}$

The graph crosses the y-axis at (0,20)and has a slope of -4. using a solid line because the inequality is <. Test the point (0,0)to see which side of the graph to shade.

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  $\begin{array}{l}12x+3y\le 60\\ 12\left(0\right)+3\left(0\right)\le 60\\ 0\le 60\end{array}$

Since 0 is less than or equal to 60, shading the half plane that contains (0,0).