Question

1. an inequality is shown.

\\(\frac{3}{5}(18 - 8k) > -12\\)
what is the solution to the inequality?
\\(\circ\\ k < \frac{19}{4}\\)
\\(\circ\\ k > \frac{10}{4}\\)
\\(\circ\\ k < \frac{15}{4}\\)
\\(\circ\\ k > \frac{1}{4}\\)

1. the graph of a two - variable linear inequality is shown.

graph of a dashed line on a coordinate plane, with axes labeled x and y, and the line passing through some points
write the inequality shown on the graph in slope - intercept form.

Explanation:

Question 29

Step1: Eliminate the fraction

Multiply both sides of the inequality $\frac{3}{5}(18 - 8k) > -12$ by $\frac{5}{3}$ to get rid of the fraction.
$$18 - 8k > -12\times\frac{5}{3}$$

Step2: Simplify the right side

Calculate $-12\times\frac{5}{3}=-20$, so the inequality becomes:
$$18 - 8k > -20$$

Step3: Subtract 18 from both sides

Subtract 18 from both sides:
$$-8k > -20 - 18$$
$$-8k > -38$$

Step4: Divide by -8 (reverse inequality)

Divide both sides by -8. Remember to reverse the inequality sign when dividing by a negative number:
$$k < \frac{-38}{-8}$$
Simplify $\frac{-38}{-8}=\frac{19}{4}$. So $k < \frac{19}{4}$.

Step1: Find the slope

The line passes through points $(0, 2)$ (y - intercept) and $(5, 0)$. The slope $m$ is calculated as $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 2}{5 - 0}=\frac{-2}{5}$.

Step2: Determine the inequality sign

The line is dashed, so the inequality is strict (either $>$ or $<$). The shaded region (assuming from the dashed line direction) is above or below? Let's check a test point, say $(0,0)$. Plug into the line equation $y = -\frac{2}{5}x + 2$. At $(0,0)$, $0 < 2$, so if the shaded region is above the line (or we check the direction). Wait, the dashed line goes from (0,2) to (5,0). Let's write the slope - intercept form of the line first: $y=-\frac{2}{5}x + 2$. Since the line is dashed and let's assume the shaded region is below the line (from the graph's dashed line and the direction), so the inequality is $y < -\frac{2}{5}x + 2$. Wait, let's re - check the points. Wait, maybe the y - intercept is 2 (from the graph, when x = 0, y = 2) and when y = 0, x = 5. So slope $m=\frac{0 - 2}{5 - 0}=-\frac{2}{5}$. The line is dashed, so the inequality is either $y < -\frac{2}{5}x + 2$ or $y > -\frac{2}{5}x + 2$. Let's take a test point, say (0,0). Plug into $y$ and the line equation: $0$ vs $-\frac{2}{5}(0)+2 = 2$. Since $0 < 2$, if the shaded region is below the line (the region containing (0,0)), then the inequality is $y < -\frac{2}{5}x + 2$.

Answer:

$k < \frac{19}{4}$ (Corresponding to the first option: $k < \frac{19}{4}$)

Question 30