Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < \frac{1}{2}x + 2$
$y > -\frac{5}{4}x - 5$

Explanation:

Step1: Find intersection of two inequalities

We have the system of inequalities:
\[

$$\begin{cases} y < \frac{1}{2}x + 2 \\ y > -\frac{5}{4}x - 5 \end{cases}$$

\]
To find a point in the solution set, we can choose a value of \(x\) and find the corresponding range of \(y\). Let's choose \(x = 0\).

Step2: Substitute \(x = 0\) into the inequalities

For the first inequality \(y < \frac{1}{2}(0)+ 2\), we get \(y < 2\).
For the second inequality \(y > -\frac{5}{4}(0)- 5\), we get \(y > - 5\).

So when \(x = 0\), \(y\) can be any value between \(-5\) and \(2\). Let's choose \(y = 0\) (since it's a simple value in that range).

Answer:

\((0,0)\) (Note: There are infinitely many solutions, and \((0,0)\) is just one example. Other points like \((4,3)\) (check: \(3<\frac{1}{2}\times4 + 2=4\) and \(3>-\frac{5}{4}\times4-5=-10\)) also work, but \((0,0)\) is a straightforward one.)