Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set. $y \geq \frac{1}{2}x + 2$ $y > -\frac{3}{2}x - 2$ graph of coordinate plane with grid answer attempt 1 out of 3 point: (\square, \square) submit answer

Explanation:

Step1: Analyze the first inequality \( y \geq \frac{1}{2}x + 2 \)

This is a linear inequality with a slope of \( \frac{1}{2} \) and a y - intercept of 2. The line is solid (because of the "greater than or equal to" sign) and we shade above the line.

Step2: Analyze the second inequality \( y > -\frac{3}{2}x - 2 \)

This is a linear inequality with a slope of \( -\frac{3}{2} \) and a y - intercept of - 2. The line is dashed (because of the "greater than" sign) and we shade above the line.

Step3: Find the intersection of the two solution regions

To find a point in the solution set, we can pick a value of \( x \) and check if there is a \( y \) that satisfies both inequalities. Let's choose \( x = 0 \).
For the first inequality, when \( x = 0 \), \( y\geq\frac{1}{2}(0)+2=2 \).
For the second inequality, when \( x = 0 \), \( y>-\frac{3}{2}(0)-2=-2 \).
A value of \( y = 3 \) (which is greater than 2 and also greater than - 2) satisfies both inequalities. So the point \( (0,3) \) is in the solution set.

Answer:

\((0, 3)\)