which linear inequality is graphed with y > -x - 2 to create the given solution set?
○ y > x + 1
○ y < x - 1
○ y > x - 1
○ y < x + 1

Question

which linear inequality is graphed with y > -x - 2 to create the given solution set? ○ y > x + 1 ○ y < x - 1 ○ y > x - 1 ○ y < x + 1

Accepted Answer

# Explanation: ## Step1: Analyze the second line's slope and intercept The general form of a linear inequality is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. Looking at the graph, the second line (the one we need to find the inequality for) has a slope of $1$ (since for a line $y=mx + b$, if we consider the rise over run, for example, from $(0,1)$ to $(1,2)$, the rise is $1$ and the run is $1$, so $m = 1$) and a y - intercept of $1$ (it crosses the y - axis at $(0,1)$). So the equation of the line is $y=x + 1$ or $y=x-1$ or other forms. But we need to check the inequality sign. ## Step2: Determine the inequality sign The solution set is the overlapping region of the two inequalities. The first inequality is $y>-x - 2$ (the region above the line $y=-x - 2$). The second line: we can test a point in the overlapping region. Let's take the point $(0,1)$. Let's check the options: - For $y>x + 1$: Plugging $x = 0,y = 1$, we get $1>0 + 1$ which is $1>1$, false. - For $yx - 1$: Plugging $x = 0,y = 1$, we get $1>0 - 1=-1$, true. But wait, let's check the region. Wait, the overlapping region (the red area) is above the line $y=-x - 2$ and also above or below the other line? Wait, no, let's look at the graph. The second line: the line passes through $(0,1)$ and has a slope of $1$. The overlapping region (the darker red) is above $y=-x - 2$ and also above $y=x - 1$? Wait, no, maybe I made a mistake. Wait, the other line: let's check the y - intercept. Wait, the line in the graph (the non - $y=-x - 2$ line) crosses the y - axis at $(0,1)$ and has a slope of $1$, so the equation is $y=x + 1$? Wait, no, when $x = 0$, $y = 1$, and when $x=1$, $y = 2$, so $y=x + 1$. But the inequality: the region of the second inequality. The overlapping region is below the line $y=x + 1$? Wait, let's test the point $(0,0)$ in the overlapping region. For $yx + 1$: $0>0 + 1$, false. For $yx - 1$: $0>0 - 1=-1$, true. Wait, but the overlapping region: the first inequality is $y>-x - 2$ (above $y=-x - 2$) and the second inequality: looking at the graph, the line $y=x + 1$ has the region below it (since the overlapping region is below $y=x + 1$ and above $y=-x - 2$). Let's check the point $(0,0)$: - For $y-x - 2$: $0>-0 - 2=-2$ is true. So the overlapping region is the set of points that satisfy both $y>-x - 2$ and $y-x - 2$: $1>-1 - 2=-3$, true. For $yx + 1$: $1>2$, false. For $yx - 1$: $1>0$, true, but $(1,1)$ in $y

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the second line's slope and intercept

Step2: Determine the inequality sign

Answer: