Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < \frac{1}{2}x + 2$
$y > -\frac{5}{4}x - 5$
(there is a coordinate axis image here)
answer
point: (\\(\square\\), \\(\square\\))

Explanation:

Step1: Analyze the first inequality

The inequality is \( y < \frac{1}{2}x + 2 \). This is a linear inequality with a slope of \( \frac{1}{2} \) and a y - intercept of 2. The line \( y=\frac{1}{2}x + 2 \) should be drawn as a dashed line (since the inequality is strict, \( y<\) not \( y\leq\)) and we shade the region below the line.

Step2: Analyze the second inequality

The inequality is \( y>-\frac{5}{4}x - 5 \). This is a linear inequality with a slope of \( -\frac{5}{4} \) and a y - intercept of - 5. The line \( y =-\frac{5}{4}x-5\) should be drawn as a dashed line (since the inequality is strict, \( y >\) not \( y\geq\)) and we shade the region above the line.

Step3: Find the intersection region

The solution set of the system of inequalities is the region that is shaded for both inequalities. To find a point in the solution set, we can pick a point that satisfies both inequalities. Let's try the point \((0,0)\).

For the first inequality: Substitute \( x = 0\) and \( y=0\) into \( y<\frac{1}{2}x + 2\). We get \( 0<\frac{1}{2}(0)+2\), which simplifies to \( 0 < 2\), this is true.

For the second inequality: Substitute \( x = 0\) and \( y = 0\) into \( y>-\frac{5}{4}x-5\). We get \( 0>-\frac{5}{4}(0)-5\), which simplifies to \( 0>- 5\), this is also true.

Answer:

Point: \((0,0)\)