Exercise 27 Page 610

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

1. Write an equation for the boundary line.
2. 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've identified two points, (- 2,0) and (1,2), 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. rise/run=2/3 ⇔ m= 2/3 Now we can partially write the equation of the boundary line. y= mx+ b ⇒ y= 2/3x+ b In our equation b is y-coordinate of the y-intercept. In order to find it, we can substitute point (1,2) for x and y in our equation. Let's do it!

y=2/3x+b

SubstituteII

x= 1, y= 2

2= 2/3( 1)+b

▼

Solve for b

MultByOne

a * 1=a

2=2/3+b

SubEqn

LHS-2/3=RHS-2/3

4/3=b

RearrangeEqn

Rearrange equation

b= 4/3

Now, 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= mx+ b ⇒ y= 2/3x+ 4/3

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 ? 2/3x+4/3 To figure out what the symbol should be, we need a test point that lies within the solution set.

We will substitute ( - 2, 2) for this test, then make the inequality symbol fit the resulting statement.

y ? 2/3x+4/3

SubstituteII

x= - 2, y= 2

2 ? 2/3( - 2)+4/3

MoveRightFacToNum

a/c* b = a* b/c

2 ? - 4/3+4/3

AddFrac

Add fractions

2 ? 0/3

0/a=0

2 ? 0

Two is greater than 0. We can infer that, of the four inequality symbols, only two would make this a true statement, > or ≥. Returning to the given graph one last time, we see that the boundary line is dashed, not solid. This implies that the inequality is strict. We can now form our final inequality. y>2/3x+4/3