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 - y \leq 4 x \geq 0 y \geq 0 (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 - y \leq 4 2 x - y = 4

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

2 x - y = 4

▼

Write in slope-intercept form

SubEqn

$LHS - 2 x = RHS - 2 x$

- y = - 2 x + 4

MultEqn

$LHS \cdot (- 1) = RHS \cdot (- 1)$

y = 2 x - 4

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 = 2 x + (- 4)

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 - y \leq 4

SubstituteII

$x = 0$ , $y = 0$

2 (0) - 0 \leq ? 4

ZeroPropMult

Zero Property of Multiplication

0 \leq 4

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

Inequality II

Now that we have completed the first inequality, let's determine the boundary line of the second inequality, using the same process.

Inequality : Boundary Line : x \geq 0 x = 0

The boundary line of this inequality is a vertical line. The inequality

x \geq 0

describes all values of

x

that are greater than or equal to

0 .

This means that every coordinate pair with an

x - value

that is greater than or equal to

0

needs to be included in the shaded region. Notice that the inequality is not strict, so the boundary line will be solid.

Inequality III

The third inequality only has a

y - variable .

This means that the boundary will be a horizontal line.

Inequality : Boundary Line : y \geq 0 y = 0

Because

y

is greater than or equal to

0,

we will shade above the boundary line. Since the inequality is not strict, the boundary line 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 three inequalities.

Exercise