Question

5x ? 2y - 24, 3x + 4y ? 8 (with a coordinate grid chart showing shaded regions and two lines)

Explanation:

To determine the inequality symbols, we analyze the lines and shaded regions:

Step 1: Analyze \( 3x + 4y \, \square \, 8 \)

• The line \( 3x + 4y = 8 \) is dashed (orange), so the inequality is strict (\( < \) or \( > \)).
• The shaded region for \( 3x + 4y \) is below the line (since the dashed line slopes downward, and the orange shade is in the lower half). Testing a point (e.g., \( (0,0) \)): \( 3(0) + 4(0) = 0 < 8 \), so \( 3x + 4y < 8 \).

Step 2: Analyze \( 5x - 2y \, \square \, 24 \)

• The line \( 5x - 2y = 24 \) is solid (blue), so the inequality is non - strict (\( \leq \) or \( \geq \)).
• The shaded region for \( 5x - 2y \) is above the line (the blue shade is in the upper half relative to the line). Rewrite \( 5x - 2y = 24 \) as \( y=\frac{5}{2}x - 12 \). Testing a point (e.g., \( (0,0) \)): \( 5(0)-2(0)=0 \), and \( 0 < 24 \), but the shaded region is above, so we need the opposite. Solving for \( y \): \( - 2y\geq24 - 5x\Rightarrow y\leq\frac{5}{2}x - 12 \), so \( 5x - 2y\geq24 \) (or \( 5x - 2y\geq24 \) is equivalent to \( - 2y\geq24 - 5x\Rightarrow y\leq\frac{5}{2}x - 12 \), and the shaded region is above the line \( 5x - 2y = 24 \) which matches \( 5x - 2y\geq24 \)).

Answer:

For \( 3x + 4y \, \square \, 8 \): \( 3x + 4y < 8 \)
For \( 5x - 2y \, \square \, 24 \): \( 5x - 2y\geq24 \) (or \( 5x - 2y > 24 \) if the line were dashed, but it is solid here, so \( \geq \))

(Note: If the blue line was intended to be dashed, it would be \( 5x - 2y > 24 \), but based on the solid line, it is \( \geq \).)