Q. 421 - Intermediate Algebra

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?

Step-by-Step Solution

Answer

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

(b) 

(c) no

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:

$$\begin{gathered} x+0.5y\le 50 \\ 20+0.5\left(40\right) \\ =40 \\ and 40\le 50 \\ Which satisfies the above inequality. \\ Now, lets check for another inequality \\ y\ge 3x \\ 40\ge 3\left(20\right) \\ 40\ge 60 \\ Which does not holds. \end{gathered}$$