Exercise 4 Page 160

To solve the given system by graphing, we will first graph each inequality separately, then combine the graphs. The overlapping region will be the solution of the system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 2 x + 3 y \leq 18 0 \leq x \leq 5 0 \leq y \leq 4 (I) (II) (III)

Let's start!

Inequality I

To graph an inequality, we first determine its boundary line. By changing the inequality symbol to an equals sign, the boundary line can be determined.

Inequality : Boundary Line : 2 x + 3 y \leq 18 2 x + 3 y = 18

Since the boundary line is not in slope-intercept form, we will rewrite it in slope-intercept form.

2 x + 3 y = 18

▼

Write in slope-intercept form

SubEqn

$LHS - 2 x = RHS - 2 x$

3 y = - 2 x + 18

DivEqn

$LHS / 3 = RHS / 3$

y = - \frac{2}{3} x + 6

Now we can determine the slope

m

and the

y -

intercept

b

of the line.

Slope - Intercept Form : Boundary Line : y = m x + b y = - \frac{2}{3} x + 6

Now that we know the slope and the

y -

intercept, we can use them to draw the boundary line. Note that the boundary line will be solid because the inequality is not strict.

By testing a point that is not on the boundary line, we can determine which region we should shade. Let's test the point

(0, 0) .

If it satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.

2 x + 3 y \leq 18

SubstituteII

$x = 0$ , $y = 0$

2 (0) + 3 (0) \leq ? 18

ZeroPropMult

Zero Property of Multiplication

0 + 0 \leq ? 18

AddTerms

Add terms

0 \leq 18

Since the test point satisfies the inequality, we will shade the region that contains the point.

Inequality II

Next inequality contains only the

x -

variable.

0 \leq x \leq 5 ⇕ x \geq 0 and x \leq 5

This compound inequality describes all values of

x

that are greater than or equal to

0

and less than or equal to

5 .

This means that all coordinate pair with an

x -

coordinate that is greater than or equal to

0

and less than or equal to

5

will be in the solution set. Note that both inequalities are non-strict, so the boundary lines will be solid.

Inequality III

We will follow a similar process for the third inequality, however, this time it has only the

y - variable .

0 \leq y \leq 4 ⇕ y \geq 0 and y \leq 4

This compound inequality describes all values of

y

that are greater than or equal to

0

and less than or equal to

4 .

Again, note that both inequalities are non-strict, so the boundary lines will be solid.

Combining the Inequality Graphs

By drawing all three inequality graphs on the same coordinate plane, we can show the overlapping section. This is the solution set of the system.

Finally, to view only the overlapping region, we remove any shaded regions that are not overlapping for all inequalities.