Question

solving systems of linear inequalities
determining linear inequalities with no solution

which linear equality will not have a shared solution set with the graphed linear inequality?
( y > \frac{2}{5}x + 2 )
( y > -\frac{2}{5}x - 5 )
( y < -\frac{5}{2}x - 7 )
( y < \frac{5}{2}x + 2 )

Explanation:

Step1: Analyze the graphed inequality

The graphed inequality is \( y \geq -\frac{5}{2}x - 3 \). The solution set is the region above (including the line) of \( y = -\frac{5}{2}x - 3 \).

Step2: Analyze each option

• Option 1: \( y > \frac{2}{5}x + 2 \). The lines \( y = -\frac{5}{2}x - 3 \) and \( y=\frac{2}{5}x + 2 \) have different slopes, so their regions may overlap.
• Option 2: \( y > -\frac{2}{5}x - 5 \). The slope is different from the graphed line, and regions may overlap.
• Option 3: \( y < -\frac{5}{2}x - 7 \). The line \( y = -\frac{5}{2}x - 7 \) is parallel to \( y = -\frac{5}{2}x - 3 \) (same slope \( -\frac{5}{2} \)) and is below it (since \( -7 < -3 \)). The region \( y < -\frac{5}{2}x - 7 \) is below this lower parallel line, while the graphed region is above \( y = -\frac{5}{2}x - 3 \). So these regions have no overlap.
• Option 4: \( y < \frac{5}{2}x + 2 \). The slope is different, and regions may overlap.

Answer:

\( y < -\frac{5}{2}x - 7 \) (the option \( y < -\frac{5}{2}x - 7 \))