Exercise 71 Page 612

It looks like given inequality is a quadratic inequality. The boundary curve given by the parabola seems to pass through the x-axis at x = -2 and x = 2 and the y-axis at y = 4. Let's recall the factored form of a quadratic function. y = a(x - x_1)(x - x_2)In equation above x_1 and x_2 are roots of the quadratic function. In our case those roots are -2 and 2. This allows us to partially complete the boundary curve equation. y = a( x - ( -2))(x - 2) ⇕ y = a(x+2)(x-2) Next, we want to find the value of the a-variable. To do so, we will use the last observation we made — the parabola crosses the y-axis at y = 4. Therefore, for x = 0 we must have y = 4. Let's substitute these values into equation above and solve for a.

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

SubstituteII

x= 0, y= 4

4 = a( 0+2)( 0-2)

AddSubTerms

Add and subtract terms

4 = a(2)(-2)

Multiply

Multiply

4 = -4 a

DivEqn

.LHS /(-4).=.RHS /(-4).

-1 = a

RearrangeEqn

Rearrange equation

a = -1

We found that a = -1. Let's complete the equation for the boundary curve using this information. y = -(x+2)(x-2) Now that we know the equation of the boundary curve, we can find the formula for the inequality. First, notice that the boundary curve is dashed, so the inequality is strict. This allows us to narrow down the inequalities to two possibilities. y > -(x+2)(x-2) y < -(x+2)(x-2) To find which of the two inequalities is described by our inequality. Notice that since we know that point (0,4) is on the boundary curve and on the diagram, all points below (0,4) satisfy this inequality. This tells us that it is the latter option which describes our inequality. y < -(x+2)(x-2) Finally, now that we know that it is the inequality above, which is depicted on the given diagram, we can finally check whether point (1,1) would make the inequality true. To do so, let's substitute it into formula above and check whether it makes the inequality true.

y < -(x+2)(x-2)

SubstituteII

x= 1, y= 1

1 ? < -( 1+2)( 1-2)

AddSubTerms

Add and subtract terms

1 ? < -(3)(-1)

Multiply

Multiply

1 < 3 ✓

Therefore, point (1,1) makes our inequality true.