Q 321 - Intermediate Algebra

Question

In the following exercises, translate to a system of inequalities and solve 

Juan is studying for his final exams in chemistry and algebra he knows he only has $24$ hours to study, and it will take him at least three times as long to study for algebra than chemistry.

ⓐ Write a system of inequalities to model this situation. 

ⓑ Graph the system. 

ⓒ Can he spend $4$ hours on chemistry and $20$ hours on algebra?

ⓓ Can he spend $6$ hours on chemistry and $18$ hours on algebra?

Step-by-Step Solution

Answer

Part a. A system of inequalities to model this situation is $\left\{\begin{array}{l}x+y\le 24\\ y\ge 3x\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part b. The graph of the system is

Part c. Yes, he can spend $4$ hours on chemistry and $20$ hours on algebra.

Part d. Yes, he can spend $6$ hours on chemistry and $18$ hours on algebra

Part (a) Step 1. Form the system of inequalities

Let $x$ represents the number of hours to study chemistry and $y$ represents the number of hours to study algebra.

He only has $24$ hours to study, so the first inequality is

$x+y\le 24 ...\left(1\right)$

It will take him at least three times as long to study for algebra than chemistry, so the second inequality can be written as

$y\ge 3x ...\left(2\right)$

As the number of hours is always non-negative, so the final system of inequalities is given as

$\left\{\begin{array}{l}x+y\le 24\\ y\ge 3x\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Graph the first inequality

Graph the inequality $x+y\le 24$ by drawing the line $x+y=24$ and testing the point.

The boundary will be solid and the point $(0,0)$ makes the inequality true, so shade black in the side that contains $(0,0)$ and so the graph is

Part (b) Step 2. Graph the second inequality on the same coordinate plane

Graph the line $y=3x$, the point $(0,5)$ makes the inequality $y\ge 3x$ true, so the shade red to the side that contains $(0,5)$ in the same plane.

The common shaded region is the solution of the system of inequality.

Part (c) Step 1. Check the point on the graph

$4$ hours on chemistry and $20$ hours on algebra represents the point $(4,20)$ on the coordinate plane, so on plotting the point on the plane we get

The point lies in the common shaded region, so it is a solution of the system of inequalities.

So, he can spend $4$ hours on chemistry and $20$ hours on algebra.

Part (d) Step 1. Check the point on the graph

$6$ hours on chemistry and $18$ hours on algebra represents the point$(6,18)$ on the coordinate plane, so on plotting the point on the plane we get

The point lies in the common shaded region, so it is a solution of the system of inequalities.

So, he can spend $6$ hours on chemistry and $18$ hours on algebra.