Question

what inequality is shown by each graph?
see example 3
24.
25.
26.
27.
or
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Explanation:

Problem 24

Step 1: Identify the line equation

The line passes through \((-2, 0)\) and \((0, -4)\). The slope \(m=\frac{-4 - 0}{0 - (-2)}=\frac{-4}{2}=-2\). Using slope - intercept form \(y = mx + b\), with \(b=-4\) (y - intercept), the line equation is \(y=-2x - 4\).

Step 2: Determine the inequality symbol

The line is solid (so the inequality is non - strict, \(\leq\) or \(\geq\)) and the shaded region is below the line. So the inequality is \(y\leq - 2x-4\).

Problem 25

Step 1: Identify the line equation

The line is horizontal? No, let's find two points. The line passes through \((-4, -2)\) and \((0,1)\)? Wait, no, looking at the graph, the dashed line: let's find the slope. Let's take two points on the dashed line. Let's say when \(x = - 4\), \(y=-2\); when \(x = 0\), \(y = 1\). The slope \(m=\frac{1-(-2)}{0 - (-4)}=\frac{3}{4}\)? Wait, no, maybe better to see the y - intercept. Wait, the dashed line: let's re - examine. Wait, the graph has a dashed line. Let's find the equation of the dashed line. Let's take two points: from the graph, the dashed line passes through \((-4,-2)\) and \((0,1)\)? Wait, no, maybe I made a mistake. Wait, the y - intercept seems to be \(y = 1\) when \(x = 0\), and when \(x=-4\), \(y=-2\). The slope \(m=\frac{1-(-2)}{0 - (-4)}=\frac{3}{4}\)? No, wait, the line in problem 25: let's check the rise over run. Wait, maybe it's a line with slope \(\frac{3}{4}\)? Wait, no, let's do it properly. Let's take two points on the dashed line. Let's say the line passes through \((-4,-2)\) and \((0,1)\). The slope \(m=\frac{1 - (-2)}{0-(-4)}=\frac{3}{4}\). The equation of the line using point - slope form. Using point \((0,1)\), \(y-1=\frac{3}{4}(x - 0)\), so \(y=\frac{3}{4}x + 1\).

Step 2: Determine the inequality symbol

The line is dashed (so the inequality is strict, \(>\) or \(<\)) and the shaded region is above the line. So the inequality is \(y>\frac{3}{4}x + 1\). Wait, no, maybe I misread the graph. Wait, the graph in 25: the dashed line, let's check the y - intercept. Wait, maybe the line is \(y=\frac{3}{4}x + 1\)? Wait, no, let's re - evaluate. Wait, the graph has a dashed line, and the shaded region is above the line. Let's take a test point, say \((0,0)\): if we plug into \(y\) and the line equation. Wait, maybe the line is \(y=\frac{3}{4}x+1\)? Wait, no, let's look at the graph again. The dashed line: let's find two points. Let's say when \(x=-4\), \(y = - 2\); when \(x = 0\), \(y=1\). So the slope \(m=\frac{1-(-2)}{0 - (-4)}=\frac{3}{4}\), equation \(y=\frac{3}{4}x + 1\). The shaded region is above the dashed line, so the inequality is \(y>\frac{3}{4}x + 1\).

Problem 26

Step 1: Identify the line equation

The dashed line: let's find two points. Let's take \((-6,0)\) and \((0,4)\). The slope \(m=\frac{4 - 0}{0-(-6)}=\frac{4}{6}=\frac{2}{3}\). Using slope - intercept form \(y=mx + b\), with \(b = 4\) (y - intercept), the line equation is \(y=\frac{2}{3}x+4\).

Step 2: Determine the inequality symbol

The line is dashed (so the inequality is strict, \(>\) or \(<\)) and the shaded region is above the line. So the inequality is \(y>\frac{2}{3}x + 4\).

Problem 27

Answer:

Step 1: Identify the line equation

The line is solid, passes through \((0,5)\) and \((6,3)\). The slope \(m=\frac{3 - 5}{6-0}=\frac{-2}{6}=-\frac{1}{3}\). Using slope - intercept form \(y=mx + b\), with \(b = 5\), the line equation is \(y=-\frac{1}{3}x + 5\).

Step 2: Determine the inequality symbol

The line is solid (so the inequality is non - strict, \(\leq\) or \(\geq\)) and the shaded region is above the line. So the inequality is \(y\geq-\frac{1}{3}x + 5\).

Problem 24 Answer: \(y\leq - 2x-4\)

Problem 25 Answer: \(y>\frac{3}{4}x + 1\) (assuming the line equation is correct)

Problem 26 Answer: \(y>\frac{2}{3}x + 4\)

Problem 27 Answer: \(y\geq-\frac{1}{3}x + 5\)

(Note: The above solutions are based on the general method of finding the equation of the line (using slope - intercept form) and then determining the inequality symbol based on the type of line (solid/dashed) and the direction of the shaded region (above/below the line). For each graph, we first find two points on the line to calculate the slope, then find the y - intercept to get the line equation, and then determine the inequality symbol by checking if the line is solid (non - strict inequality) or dashed (strict inequality) and the position of the shaded region relative to the line.)