Questions: Graph the solution set. x-y ≥ 3 x+y ≤ 3

Graph the solution set. x-y ≥ 3 x+y ≤ 3
Transcript text: Graph the solution set. \[ \begin{array}{l} x-y \geq 3 \\ x+y \leq 3 \end{array} \]
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Solution

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Solution Steps

Step 1: Rewrite the Inequalities in Slope-Intercept Form

The given inequalities are: \[ x - y \geq 3 \] \[ x + y \leq 3 \]

Rewrite them in slope-intercept form (\(y = mx + b\)):

For \( x - y \geq 3 \): \[ -y \geq -x + 3 \] \[ y \leq x - 3 \]

For \( x + y \leq 3 \): \[ y \leq -x + 3 \]

Step 2: Graph the Boundary Lines

Graph the lines \( y = x - 3 \) and \( y = -x + 3 \) on the coordinate plane. These lines will be the boundaries for the inequalities.

Step 3: Determine the Shaded Regions

For \( y \leq x - 3 \), shade the region below the line \( y = x - 3 \).

For \( y \leq -x + 3 \), shade the region below the line \( y = -x + 3 \).

Step 4: Identify the Solution Set

The solution set is the region where the shaded areas overlap. This is the area that satisfies both inequalities.

Final Answer

The solution set is the region below both lines \( y = x - 3 \) and \( y = -x + 3 \). This region is a triangular area bounded by the lines and the x-axis.

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