Question

question
solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y < \frac{3}{2}x - 6$

Explanation:

Step1: Analyze the inequality

The inequality is \( y < \frac{3}{2}x - 6 \). This is a linear inequality in two variables. The boundary line is \( y=\frac{3}{2}x - 6 \), which has a slope of \( \frac{3}{2} \) and a y - intercept of \( - 6 \). Since the inequality is \( y<\frac{3}{2}x - 6 \), the boundary line should be dashed (because the inequality is strict, \( y\) is not equal to \( \frac{3}{2}x - 6 \)) and we shade the region below the line.

Step2: Find a test point

To find a point in the solution set, we can choose a point that satisfies the inequality. Let's choose \( x = 4 \). Substitute \( x = 4 \) into the right - hand side of the inequality: \( y<\frac{3}{2}(4)-6=\frac{12}{2}-6 = 6 - 6=0 \). Let's choose \( y=-1 \). Check if \( - 1<\frac{3}{2}(4)-6 \). Since \( \frac{3}{2}(4)-6 = 0 \) and \( - 1 < 0 \), the point \( (4,-1) \) satisfies the inequality \( y < \frac{3}{2}x - 6 \). We can also choose other points. For example, when \( x = 6 \), \( y=\frac{3}{2}(6)-6=9 - 6 = 3 \). A point like \( (6,2) \) also works because \( 2<3 \).

Answer:

One point in the solution set is \( (4,-1) \) (other valid points like \( (6,2) \) etc. are also correct).