Chapter 5.7, Problem 35E

To determine

To find: the amount that can be spend on savings and housing.

Answer to Problem 35E

The amounts that can be spent on savings and housing are

  $300\text{ and }500$ dollar.

Explanation of Solution

Given:

Spend less than half of the monthly $\$2000$ .

At least $10\%$ on savings and at most $30\%$ on housing.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but the are of the coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

The common area is the solution of inequality.

Calculation:

Th amount spend on savings and housing is less than half of $2000$ dollars.

The amount spent on savings is at least $10\%$ of $2000$ dollars.

The amount spent on housing is at most $30\%$ of $2000$ dollars.

Write the system of linear inequalities that models the given problem.

Let $x$ be the amount spent on savings and $y$ be the amount spent on housing. Then the system of inequalities that model the given problem is”

  $\begin{array}{l}x\ge 200.\\ y\le 200.\\ x+y<1000.\end{array}$

Graph the inequalities:

  $x\ge 200,y\le 200\text{ and }x+y<1000$ .

As shown in the below figure. The intersection of pink, blue and green colors represent the solution set of given problem.

For example:

The ordered pair:

  $\left(x,y\right)=\left(300,500\right)$ dollar is a solution.

It means $300$ dollars can be sent on saving and $500$ dollars can be spent on housing.

The graph of the combination inequalities:

Therefore, the amounts that can be spent on savings and housing are:

  $300\text{ and }500$ dollar.

Hence, the amounts that can be spent on savings and housing are

  $300\text{ and }500$ dollar.