Chapter 5.6, Problem 32PPS

To determine

Graph $2x-3y\le 1$ and determine which of the ordered pairs in $\{(2,3),(3,1),(0,0),\left(0,-1\right),(5,3)\}$ are in the solution set.

Answer to Problem 32PPS

Graph of inequality $2x-3y\le 1.$

The only order pairs (2,3), (0,0) and (5,3) are in the solution set.

Explanation of Solution

Calculation: Change given inequality in slope-intercept form $y=mx+b$ first, where m is the slope and b is the y -intercept.

  $2x-3y\le 1$

Subtracting 2x on both sides

  $2x-3y-2x\le 1-2x$

Simplify

  $-3y\le 1-2x$

Dividing both sides by − 3.

  $\frac{-3y}{-3}\le \frac{1}{-3}-\frac{2x}{(-3)}$

Simplify

  $y\ge \frac{2x}{3}-\frac{1}{3}$

The slope of the line $y\ge \frac{2x}{3}-\frac{1}{3}$ is $\frac{2}{3}$ and the y −intercept is $-\frac{1}{3}=0.333$ .

Plot y-intercept $\left(0,-0.333\right)$ first.

Plot one more point using $\frac{Rise}{Run}=\frac{2}{3}.$

Another coordinate will be $\left(0+3,-0.333+2\right)=(3,1.667).$

Plot points $\left(0,-0.333\right)$ and $(3,1.667)$ .

Join the points and draw a solid line for inequality sign $\ge$ and shade up the line.

Now check $(2,3),(3,1),(0,0),\left(0,-1\right),(5,3)$ order pairs on the coordinate axes.

We could see that only order pair s (2,3), (0,0) and (5,3) are inside shaded region.

Therefore, the order pair s (2,3), (0,0) and (5,3) are in the solution set.