Question

choose all that correctly graph the given inequality.
a
$y \geq -5x - 3$

b
$y \leq 3x + 2$

c
$y \geq \frac{1}{2}x + 1$

d
$y \leq \frac{1}{5}x + 2$

Explanation:

To determine which graphs correctly represent the given inequalities, we analyze each option based on the properties of linear inequalities (slope, y - intercept, and the direction of the shaded region):

Option A: \(y\geq - 5x - 3\)

1. Slope and y - intercept: The equation of the line is in slope - intercept form \(y=mx + b\), where \(m=-5\) (slope) and \(b = - 3\) (y - intercept).
2. Type of line and shading: Since the inequality is \(y\geq - 5x - 3\), the line should be solid (because of the "greater than or equal to" sign) and we shade above the line.

• Let's check a test point, say \((0,0)\). Substitute into the inequality: \(0\geq-5(0)-3\), which simplifies to \(0\geq - 3\), a true statement. The graph in option A has a solid line (correct for \(\geq\)) and the shading includes \((0,0)\), so option A is correct.

Option B: \(y\leq3x + 2\)

1. Slope and y - intercept: For the line \(y = 3x+2\), the slope \(m = 3\) and the y - intercept \(b = 2\).
2. Type of line and shading: The inequality is \(y\leq3x + 2\), so the line should be solid and we shade below the line.

• Let's take the test point \((0,0)\). Substitute into the inequality: \(0\leq3(0)+2\), which is \(0\leq2\), a true statement. The graph in option B has a solid line and the shading includes \((0,0)\) (shading below the line), so option B is correct.

Option C: \(y\geq\frac{1}{2}x + 1\)

1. Slope and y - intercept: The line \(y=\frac{1}{2}x + 1\) has a slope \(m=\frac{1}{2}\) and a y - intercept \(b = 1\).
2. Type of line and shading: The inequality is \(y\geq\frac{1}{2}x + 1\), so the line should be solid and we shade above the line.

• Let's check the test point \((0,0)\). Substitute into the inequality: \(0\geq\frac{1}{2}(0)+1\), which is \(0\geq1\), a false statement. The graph in option C shades above the line but \((0,0)\) is in the shaded region, which is incorrect. So option C is wrong.

Option D: \(y\leq\frac{1}{5}x + 2\)

1. Slope and y - intercept: For the line \(y=\frac{1}{5}x + 2\), the slope \(m=\frac{1}{5}\) and the y - intercept \(b = 2\).
2. Type of line and shading: The inequality is \(y\leq\frac{1}{5}x + 2\), so the line should be solid and we shade below the line.

• Let's take the test point \((0,0)\). Substitute into the inequality: \(0\leq\frac{1}{5}(0)+2\), which is \(0\leq2\), a true statement. The graph in option D has a solid line and the shading is below the line (includes \((0,0)\)), so option D is correct.

Answer:

A. \(y\geq - 5x - 3\) (correct graph), B. \(y\leq3x + 2\) (correct graph), D. \(y\leq\frac{1}{5}x + 2\) (correct graph)