Question

Lupe’s allowance for the month is \(60. She wants to go to a concert for which a ticket costs \)45.

1. Write and solve an inequality that shows how much money she can spend that month after buying a concert ticket.
2. She spends \(9.99 on music downloads and \)2 on lunch in the cafeteria. Write and solve an inequality that shows how much she can spend after these purchases and the concert ticket.

Short answer

1. Lupe can spend that month after buying a concert ticket no more than $15.
2. Lupe can spend no more than $3.01 for both these purchases and the concert ticket.

Step-by-step solution

Part a. Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < or >, use the open parenthesis or a hollow circle at that endpoint.

To graph the endpoint with inequality symbol $\le \text{or}\ge$, use the bracket or a solid circle at that endpoint.

Properties of inequality:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ 2.b\le c\Rightarrow ab\le ac,\text{if}a>0 \\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0 \end{gathered}$$

Part a.  Step 2. State the inequality from the given information.

Let m represents how much money she can spend that month after buying a concert ticket.

The inequality:

$m+45\le 60$

Part a.  Step 3. Solve the inequality.

Solve for m:

$$\begin{gathered} m+45\le 60 \\ m+45-45\le 60-45 \\ m\le 15 \end{gathered}$$

Lupe can spend that month after buying a concert ticket no more than $15.

Part b. Step 1. State the concept for inequality.

To graph the endpoint with strict inequality < or >, use the open parenthesis or a hollow circle at that endpoint.

To graph the endpoint with inequality symbol $\le \text{or}\ge$, use the bracket or a solid circle at that endpoint.

Properties of inequality:

$$\begin{gathered} 1.b\le c\Rightarrow b\pm a\le c\pm a \\ 2.b\le c\Rightarrow ab\le ac,\text{if}a>0 \\ 3.b\le c\Rightarrow ab\ge ac,\text{if}a<0 \end{gathered}$$

Part b. Step 2. State the inequality from the given information.

Let m represents how much money she can spend both these purchases and the concert ticket.

The inequality:

$m+45+9.99+2\le 60$

Part b. Step 3. Solve the inequality.

Solve for m:

$$\begin{gathered} m+45+9.99+2\le 60 \\ m+56.99\le 60 \\ m+56.99-56.99\le 60-56.99 \\ m\le 3.01 \end{gathered}$$

Thus, Lupe can spend no more than $3.01 for both these purchases and the concert ticket.