Question

Mrs Jones is buying new books and puzzles for her preschool classroom. Each book costs \(6, and each puzzle costs \)4. Write and graph an inequality to determine how many books and puzzles she can buy for $96.

Short answer

The inequality for the given problem is:$\mathbf{3}\mathit{x}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\le }\mathbf{48}$

The graph of the inequality$3x+2y\le 48$ is:

Mrs Jones can buy 14 books and 3 puzzles for $96.

Step-by-step solution

Step 1. Write the inequality for the given problem.

Let $x$ be the number of books.

It is given that the cost of 1 book is $6.

Therefore, the cost of$x$ books is $\$\left(6x\right)$.

Let$y$ be the number of puzzles.

It is given that the cost of 1 puzzle is $4.

Therefore, the cost of$y$ puzzles is $\$\left(4y\right)$.

It is given that Mrs Jones has a sum of money $96.

Therefore, at most Mrs Jones can spend $96.

Therefore, the sum of costs of books and puzzles is at most $96.

Therefore, the inequality for the given problem is:

$\begin{array}{c}6x+4y\le 96\\ 2\left(3x+2y\right)\le 96\\ \frac{2\left(3x+2y\right)}{2}\le \frac{96}{2}\\ 3x+2y\le 48\end{array}$

Step 2. Write the procedure to draw the graph of the inequality 3x+2y≤48.

Convert the inequality$3x+2y\le 48$ into equality that is convert the inequality into equation.

Therefore, it is obtained that:$3x+2y=48$

Therefore, the equation of the boundary is $3x+2y=48$. The inequality$3x+2y\le 48$ has equal to sign , therefore the boundary is included in the solution. Therefore, the boundary is denoted by solid lines.

Draw the graph of the boundary line $3x+2y=48$.

Substitute 0 for$x$ and find the value of $y$.

$\begin{array}{c}3x+2y=48\\ 3\left(0\right)+2y=48\\ 2y=48\\ y=\frac{48}{2}\\ y=24\end{array}$

Therefore, one of the point is

Substitute 0 for$y$ and find the value of $x$.

$\begin{array}{c}3x+2y=48\\ 3x+2\left(0\right)=48\\ 3x=48\\ x=\frac{48}{3}\\ x=16\end{array}$

Therefore the other point is$\left(16,0\right)$

Therefore, draw the graph of the boundary line$3x+2y=48$ by drawing a line passing through the points$\left(0,24\right)$ and $\left(16,0\right)$.

Now to draw the graph of the inequality $3x+2y\le 48$, take any point which is not on the line$3x+2y=48$ in the inequality $3x+2y\le 48$. If the condition obtained is true, then shade the region towards that point and if the condition obtained is false, then shade the region away from the point.

Let the point be$\left(0,0\right)$ and the point$\left(0,0\right)$ is not on the line $3x+2y=48$.

Substitute the point in the inequality $3x+2y\le 48$.

$\begin{array}{c}3x+2y\le 48\\ 3\left(0\right)+2\left(0\right)\le 48\\ 0+0\le 48\\ 0\le 48\end{array}$

As, the condition obtained is $0\le 48$, which is true. Therefore, to draw the graph of the inequality $3x+2y\le 48$, shade the region towards the point $\left(0,0\right)$.

Step 3. Draw the graph of the given inequality 3x+2y≤48 by using the above facts.

The graph of the given inequality$3x+2y\le 48$ is:

The shaded region represents the solution set of the inequality $3x+2y\le 48$.

From the graph, it can be noticed that the ordered pair $\left(14,3\right)$is in the solution set of $3x+2y\le 48$.

Therefore $x=14$, and$y=3$ is the solution of $3x+2y\le 48$.

Therefore, Mrs Jones can buy 14 books and 3 puzzles for $96.