Question

\\y > -3x - 3\\
\\y > x - 7\\

Explanation:

Identify the boundary lines

The system of inequalities is:

$$
LATEXBLOCK0
$$

The boundary lines are \(y = -3x - 3\) and \(y = x - 7\). Both inequalities use strict inequality (\(>\)), so both boundary lines must be dashed.

Find key points for the first line

For the line \(y = -3x - 3\):

• The \(y\)-intercept is \((0, -3)\).
• If \(x = -1\), \(y = 0\), giving the \(x\)-intercept \((-1, 0)\).
• If \(x = -2\), \(y = 3\).

Find key points for the second line

For the line \(y = x - 7\):

• The \(y\)-intercept is \((0, -7)\).
• If \(x = 7\), \(y = 0\), giving the \(x\)-intercept \((7, 0)\).
• If \(x = 2\), \(y = -5\).

Determine the intersection point

Set the two equations equal to find where the boundary lines intersect:

$$
LATEXBLOCK1
$$

Substitute \(x = 1\) into \(y = x - 7\):

$$ y = 1 - 7 = -6 $$

The intersection point is \((1, -6)\).

Determine the shaded region

For both inequalities, \(y\) is greater than the expression:

• For \(y > -3x - 3\), shade the region above the line \(y = -3x - 3\).
• For \(y > x - 7\), shade the region above the line \(y = x - 7\).

The solution set is the overlapping region that lies above both dashed boundary lines.

Answer:

To graph the system of linear inequalities:

1. Graph the first boundary line \(y = -3x - 3\):

• Draw a dashed line passing through \((0, -3)\) and \((-1, 0)\).
• Shade the region above this line.

1. Graph the second boundary line \(y = x - 7\):

• Draw a dashed line passing through \((0, -7)\) and \((7, 0)\).
• Shade the region above this line.

1. Identify the solution set:

• The final solution is the region where the two shaded areas overlap, which is the region located above both dashed lines, starting from their intersection point at \((1, -6)\).