Exercise 25 Page 414

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.

In this exercise, we have been given a system consisting of two linear inequalities. We will tackle them one at a time and bring them together in a system at the end.

The Yellow Region

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

Here, we've identified two points, (-5, 4) 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=-6/2 ⇔ m= -3 We can select one of those points, for example (- 1,-2). With slope m and a point at ( - 1, - 2), we can write an equation for the boundary line in point-slope form. y- y_2= m(x- x_1) ⇒ y- ( - 2)= -3(x-( -1)) We can simplify this formula to find the slope-intercept form for the equation.

y-(-2)=-3(x-(-1))

SubNeg

a-(- b)=a+b

y+2=-3(x+1)

Distr

Distribute (-3)

y+2=-3x+(-3)

AddNeg

a+(- b)=a-b

y+2=-3 x-3

SubEqn

LHS-2=RHS-2

y=-3 x-5

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 ? -3x+(-5) ⇒ y ? -3x-5 To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation.

We will substitute ( 1, 1) for this test, then make the inequality symbol fit the resulting statement.

y ? -3x-5

SubstituteII

x= 1, y= 1

1 ? -3( 1)-5

Multiply

1 ? -3 -5

SubTerms

Subtract terms

1 ? -8

1 is greater than -8, so the symbol will be either > or ≥. Since the boundary line in the given graph is dashed the inequality is strict, and we can form the first inequality in the system. y > -3x-5

The Blue Region

Writing the inequality for this region of the graph will involve the same steps as above. We will start by identifying two points.

Again, we have denoted the rise and run of the graph, giving the slope m. rise/run=4/1 ⇔ m= 4 Since we chose the y-intercept at the point (0, 3) as one of our points for this boundary line as well, we can write its equation. y= mx+ b ⇒ y= 4x+ 3 Once more, we replace the equals sign with a blank space. y ? 4x+3 We will need another point that lies within the solution set to determine the sign of this inequality.

We will substitute ( 1, 1) for this test, then make the inequality symbol fit the resulting statement.

y ? 4x+3

SubstituteII

x= 1, y= 1

1 ? 4( 1)+3

Multiply

1 ? 4+3

AddTerms

Add terms

1 ? 7

1 is less than 7, so the symbol will be either < or ≤. Since the boundary line in the given graph is solid, the inequality is not strict, and we can form the second inequality in the system. y≤ 4x+3

Writing the System

To complete the system of inequalities, we will bring both of our inequalities together in system notation. y > -3x-5 y≤ 4x+3

Hints

Check the answer

Practice exercises

Asses your skill

The Yellow Region

The Blue Region

Writing the System