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- Write an equation for the boundary line
- Determine the inequality symbol and complete the inequality

In this exercise, we have been given a system consisting of two linear inequalities. We'll tackle them one at a time and bring them together in a system at the end. Let's call the inequalities $A$ and $B.$

It only takes two points to create a unique equation for any line, so let's start by identifying two points on the boundary line.

Here we've identified two points, $(1,0)$ and $(0,2),$ and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph. We can use this information to find the slope, $m.$ $m=runrise =1-2 =-2 $ One of the points we selected, $(0,2),$ is also the $y$-intercept. With slope $m$ and the $y$-intercept a we can write an equation for the boundary line in slope-intercept form. $y=mx+b→y=-2x+2 $ Now that we found the boundary line we should find the correct inequality symbol. $y? -2x+2 $ To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation.

We'll substitute $(1,1)$ for this test, then make the inequality symbol fit the resulting statement.$y? -2x+2$

$1? -2⋅1+2$

$1? 0$

Writing the inequality for this region of the graph will involve the same steps as above. We'll start by identifying two points.

Again, we have denoted the rise and run of the graph, giving the slope $m.$ $m=runrise =13 =3 $ Since we chose the $y$-intercept at the point $(0,-2)$ as one of our points for this boundary line as well, we can write its equation. $y=3x−2 $ Once more, we replace the equality sign with a question mark. $y? 3x−2 $ To determine the correct inequaality symbol, we will use a test point.

We'll substitute $(1,0)$ for this test, then make the inequality symbol fit the resulting statement.$y? 3x−2$

$0? 3⋅1−2$

$0? 1$

b

First we need to identify two points on the line where one preferably is the $y$-intercept.

Here we've identified two points, $(2,0)$ and $(0,3),$ and indicated the horizontal and vertical changes between them. This gives us the rise and run of the graph, which will give us the slope, $m.$ $m=runrise =2-3 =-1.5 $ Now that we have the slope and the $y$-intercept we can write an equation of the line in slope-intercept form. $y=-1.5x+3$ We should now determine the correct inequality symbol. This is done using a test point. Let's use the origin.

We'll substitute $(0,0)$ for this test, then make the inequality symbol fit the resulting statement. Since $0$ is less than $3,$ the symbol will be either $<$ or $≤.$ The boundary line in the given graph is dashed, indicating that the inequality is strict. Thus, the correct symbol is $<.$ $y<-1.5x+3 $The boundary line of the second region of the graph is a vertical line. Therefore, writing an inequality for this region requires a different approach.

Notice that every point on the boundary line will have a $x$-cordinate equal to $1.$ This information is enough to write corresponding equation.
$x=1$
Since the region to the left of the line is shaded it represents all $x$-values **less than** $1.$
$x<1$
We can now write our system of inequalities.
${y<-1.5x+3x<1 $