## Precalculus (6th Edition) Blitzer

Let us consider the given inequalities \begin{align} & 3x+y\le 9 \\ & 2x+3y\ge 6 \\ & x\ge 0 \\ & y\ge 0 \end{align} Put the equals symbol in place of the inequality and rewrite the equation as given below: By finding any two solutions of the linear equation, plot the graph of the linear equation $3x+y=9$: To find the value of the x-intercept, put the value of y = 0 as given below: \begin{align} & 3x+0=9 \\ & x=\frac{9}{3} \\ & x=3 \end{align} To find the value of the y-intercept, put the value of x = 0 as given below: \begin{align} & 3\left( 0 \right)+y=9 \\ & y=9 \end{align} Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,9 \right)$ and draw a solid line passing through these points. In the inequality there is the $\le$ symbol in which the equality is included. Now, this solid line divides the plane in three regions -- the line itself and two half-planes. Now, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: Check if the test point satisfies the inequality. \begin{align} 3x+y\le 9 & \\ 3\left( 0 \right)+0\overset{?}{\mathop{\le }}\,9 & \\ 0\le 9 & \\ \end{align} Since the test point satisfies the inequality, shade the half-plane containing that test point towards the origin. Plot the graph using the intercepts as given below: By finding any two solutions of the linear equation, plot the graph of the linear equation $2x+3y\ge 6$: To find the value of the x-intercept, put the value of y = 0 as given below: \begin{align} & 2x+3\left( 0 \right)=6 \\ & 2x=6 \\ & x=\frac{6}{2} \\ & x=3 \end{align} To find the value of the y-intercept, put the value of x = 0 as given below: \begin{align} & 2\left( 0 \right)+3y=6 \\ & 3y=6 \\ & y=\frac{6}{3} \\ & y=2 \end{align} Plot the intercepts $\left( 3,0 \right)\text{ and }\left( 0,2 \right)$ and draw a solid line passing through these points; since the inequality contains the $\le$ symbol, the equality is included. Now, this dashed line divides the plane in 3 region: the line itself, and the two half planes. Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade: Check if the test point satisfies the inequality. \begin{align} & 2x+3y\ge 6 \\ & 0+0\overset{?}{\mathop{>}}\,9 \\ & 0<9 \\ \end{align} Since, the test point does not satisfy the inequality, shade the half-plane not containing that test point that is away from the origin. And to plot the equation $x\ge 0$, graph the line $x=0$ and shade the right part of the line; that is, shade the region for which x values will be positive, as the inequality contains the $\ge$ symbol. Similarly, to plot the inequality $y\ge 0$, graph the line $y=0$ and shade the right part above the line; that is, shade the region for which y values will be positive, as the inequality contains the $\ge$ symbol. See the final graph below.