Exercise 52 Page 212

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

1. Solve the related quadratic equation.
2. Plot the solutions on a number line if exist.
3. 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.

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.

Since x=0 did not produce a true statement, no x-value satisfies the inequality. The solution set is an empty set. { } or ∅