Question

solve the following system of inequalities graphically on the set of axes below. state the coordinates of a point in the solution set.
$y \geq 2x - 4$
$y > -x + 8$

Explanation:

Step1: Analyze the first inequality \( y \geq 2x - 4 \)

This is a linear inequality. The boundary line is \( y = 2x - 4 \), which has a slope of 2 and a y - intercept of - 4. Since the inequality is \( \geq \), the line is solid, and we shade above the line.

Step2: Analyze the second inequality \( y > -x + 8 \)

The boundary line is \( y=-x + 8 \), with a slope of - 1 and a y - intercept of 8. Since the inequality is \(> \), the line is dashed, and we shade above the line.

Step3: Find the intersection of the two shaded regions

To find a point in the solution set, we can pick a point and check if it satisfies both inequalities. Let's try the point \( (3,6) \).

• For \( y \geq 2x - 4 \): Substitute \( x = 3 \) and \( y = 6 \). \( 6\geq2(3)-4=6 - 4 = 2 \). Since \( 6\geq2 \), this inequality is satisfied.
• For \( y > -x + 8 \): Substitute \( x = 3 \) and \( y = 6 \). \( 6>-3 + 8=5 \). Since \( 6 > 5 \), this inequality is also satisfied.

Answer:

A point in the solution set is \((3,6)\) (other valid points like \((4,6)\), \((2,6)\) etc. are also correct as long as they satisfy both inequalities)