Questions: Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. y ≥ -x - 7 y < 3x - 3

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

y ≥ -x - 7
y < 3x - 3

Transcript text: Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. \[ \begin{array}{l} y \geq-x-7 \\ y<3 x-3 \end{array} \]

Solution

Solution Steps

Step 1: Graph the First Inequality

Start with the inequality \( y \geq -x - 7 \).
Rewrite it as the equation \( y = -x - 7 \) to find the boundary line.
Plot the line \( y = -x - 7 \) on the graph. This line has a slope of \(-1\) and a y-intercept of \(-7\).
Since the inequality is \( y \geq -x - 7 \), shade the region above the line, including the line itself, as it is a "greater than or equal to" inequality.

Step 2: Graph the Second Inequality

Consider the inequality \( y < 3x - 3 \).
Rewrite it as the equation \( y = 3x - 3 \) to find the boundary line.
Plot the line \( y = 3x - 3 \) on the graph. This line has a slope of \(3\) and a y-intercept of \(-3\).
Since the inequality is \( y < 3x - 3 \), shade the region below the line, not including the line itself, as it is a "less than" inequality.

Step 3: Identify the Solution Set

The solution set is the region where the shaded areas from both inequalities overlap.
Choose a point within this overlapping region to verify it satisfies both inequalities.
State the coordinates of this point as a solution to the system of inequalities.