Question

Graph $\mathit{y}\mathbf{>}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{5}$. Then determine which of the ordered pairs in$\left\{\left(-2,0\right),\left(-1,5\right),\left(2,3\right),\left(7,3\right)\right\}$ are in the solution set.

Short answer

The graph of the given inequality$y>-2x+5$is:

The ordered pairs$\left(2,3\right)$ and$\left(7,3\right)$ are in the solution set.

Step-by-step solution

Step 1. Write the procedure to draw the graph of the inequality y>−2x+5.

Convert the inequality $y>-2x+5$into equality that is convert the inequality into equation.

Therefore, it is obtained that: $y=-2x+5$

Therefore, the equation of the boundary is $y=-2x+5$. The inequality$y>-2x+5$ has no equal to sign , therefore the boundary is not included in the solution. Therefore, the boundary is denoted by dashed lines.

Draw the graph of the boundary line $y=-2x+5$.

Substitute 0 for$x$ and find the value of $y$.

$\begin{array}{c}y=-2x+5\\ y=-2\left(0\right)+5\\ y=0+5\\ y=5\end{array}$

Therefore, one of the point is$\left(0,5\right)$

Substitute 0 for$y$ and find the value of $x$.

$\begin{array}{c}y=-2x+5\\ 0=-2x+5\\ 2x=5\\ x=\frac{5}{2}\\ x=2.5\end{array}$

Therefore the other point is$\left(2.5,0\right)$

Therefore, draw the graph of the boundary line$y=-2x+5$ by drawing a line passing through the points$\left(0,5\right)$ and $\left(2.5,0\right)$.

Now to draw the graph of the inequality $y>-2x+5$, take any point which is not on the line$y=-2x+5$ in the inequality $y>-2x+5$. If the condition obtained is true, then shade the region towards that point and if the condition obtained is false, then shade the region away from the point.

Let the point be$\left(0,0\right)$ and the point$\left(0,0\right)$ is not on the line $y=-2x+5$.

Substitute the point in the inequality $y>-2x+5$.

$\begin{array}{c}y>-2x+5\\ 0>-2\left(0\right)+5\\ 0>0+5\\ 0>5\end{array}$

As, the condition obtained is $0>5$, which is false. Therefore, to draw the graph of the inequality $y>-2x+5$, shade the region away from the point $\left(0,0\right)$.

Step 2. Draw the graph of the given inequality y>−2x+5 by using the above facts.

The graph of the given inequality$y>-2x+5$ is:

The shaded region represents the solution set of the inequality $y>-2x+5$.

Step 2. Determine which of the ordered pairs in {−2,0,−1,5,2,3,7,3} are in the solution set.

From the graph of the inequality $y>-2x+5$, it can be noticed that$\left(-2,0\right)$ and$\left(-1,5\right)$ lies outside the shaded region. Therefore, the ordered pairs$\left(-2,0\right)$ and$\left(-1,5\right)$ are not in the solution set.

From the graph of the inequality $y>-2x+5$, it can be noticed that$\left(2,3\right)$ and$\left(7,3\right)$ lies inside the shaded region. Therefore, the ordered pairs$\left(2,3\right)$ and$\left(7,3\right)$ are in the solution set.