Question

Translate to a system of

inequalities and solve.

Andi wants to spend no more than

\(50 on Halloween treats. She

wants to buy candy bars that cost

\)1 each and lollipops that cost

$0.50 each, and she wants the

number of lollipops to be at least

three times the number of candy

bars.

(a) Write a system of inequalities to

model this situation.

(b) Graph the system.

(c) Can she buy 20 candy bars and

40 lollipops?

Short answer

(a)$$\begin{gathered} x\ge 0 \\ y\ge 0 \\ x+0.5y\le 50 \\ y\ge 3x \end{gathered}$$

(b)

(c) no

Step-by-step solution

Part (a) Step 1. Given information.

Cost of candy bar $\$1$.

Cost of lollipops $\$0.05$.

Money to be spent $\le 50\$$

Number of lollipops is three times the number of candy bars.

Part (a) Step 2. Framing equations.

From problem given above following equations can be drawn:

$$\begin{gathered} x\ge 0 \\ y\ge 0 \\ x+0.5y\le 50......\left(1\right) \\ y\ge 3x.........\left(2\right) \end{gathered}$$

Part (b) Step 1. Given information.

$$\begin{gathered} x\ge 0 \\ y\ge 0 \\ x+0.5y\le 50 \\ y\ge 3x \end{gathered}$$

Part (b) Step 2. Drawing the graph.

Constructing the graph using given data:

$$\begin{gathered} x\ge 0 \\ y\ge 0 \\ x+0.5y\le 50 \\ y\ge 3x \end{gathered}$$

Part (c) Step 1. Given information.

$$\begin{gathered} x\ge 0 \\ y\ge 0 \\ x+0.5y\le 50 \\ y\ge 3x \end{gathered}$$

No of candy bars = 20

No of Lollipops = 40

Part (c) Step 2. Calculation.

Substituting the values of candy bars and lollipops we get:

$$\begin{gathered} x+0.5y\le 50 \\ 20+0.5\left(40\right) \\ =40 \\ and40\le 50 \\ Whichsatisfiestheaboveinequality. \\ Now,letscheckforanotherinequality \\ y\ge 3x \\ 40\ge 3\left(20\right) \\ 40\ge 60 \\ Whichdoesnotholds. \end{gathered}$$