Exercise 52 Page 212

To solve the quadratic inequality algebraically, we will follow three steps.

Solve the related quadratic equation.
Plot the solutions on a number line if exist.
Test a value to see if it satisfies the original inequality.

Step 1

We will start by solving the related equation. x^2-4x=- 7 ⇔ 1x^2+( - 4)x+ 7=0 We see above that a= 1, b= - 4, and c= 7. Let's substitute these values into the Quadratic Formula to solve the equation.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- ( - 4)±sqrt(( - 4)^2-4( 1)( 7))/2( 1)

▼

Simplify right-hand side

NegNeg

- (- a)=a

x=4±sqrt((- 4)^2-4(1)(7))/2(1)

CalcPow

Calculate power

x=4±sqrt(16-4(1)(7))/2(1)

Multiply

x=4±sqrt(16-28)/2

SubTerm

Subtract term

x=4±sqrt(- 12)/2

We have found a negative discriminant, - 12. Therefore, the related equation has no real solutions.

Step 2

Since the related equation does not have any real solution, we do not have any point to plot on a number line.We have two cases, either all x-values satisfy the original inequality or no x-value satisfies it.

Step 3

Finally, we must test a value to see if it satisfies the original inequality. Testing one value will help us to determine the solution set. For simplicity, we will choose x=0.

x^2-4x ≤ - 7

Substitute

x= 0

( 0)^2-4( 0)? ≤ - 7

▼

Simplify left-hand side

CalcPow

Calculate power

0-4(0)? ≤ - 7

ZeroPropMult

Zero Property of Multiplication

0-0? ≤ - 7