Exercise 11 Page 331

There are two major steps to writing an inequality when given its graph.

Write an equation for the boundary line.
Determine the inequality symbol and complete the inequality.

Let's get started by focusing on the boundary line.

Writing the Boundary Line Equation

It only takes two points to create a unique equation for any line, so let's identify two points on the boundary line.

Here we have identified two points,

(0, 1)

and

(3, - 1),

and indicated the horizontal and vertical changes between them. This gives us the

rise

and

run

of the graph, which will give us the slope

m .

\frac{rise}{run} = \frac{- 2}{3} \Leftrightarrow m = - \frac{2}{3}

One of the points we selected,

(0, 1),

is also the

y -

intercept, which is great news! This means that we can combine the slope

m

and the

y -

intercept at the point

(0, b)

to write an equation for the boundary line in slope-intercept form.

y = m x + b \Rightarrow y = - \frac{2}{3} x + 1

Forming the Inequality

To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space, since it is still unknown to us.

y ? - \frac{2}{3} x + 1

To figure out what the symbol should be, we need a test point that lies within the solution set.

We will substitute

(0, 0)

for this test, then make the inequality symbol fit the resulting statement.

y ? - \frac{2}{3} x + 1

SubstituteII

$x = 0$ , $y = 0$

0 ? - \frac{2}{3} (0) + 1

ZeroPropMult

Zero Property of Multiplication

0 ? 1

Zero is less than

1 .

We can infer that, of the four inequality symbols, only two would make this a true statement,

<

or

\leq .

Returning to the given graph one last time, we see that the boundary line is solid, not dashed. This implies that the inequality is not strict. We can now form our final inequality.

y \leq - \frac{2}{3} x + 1