Exercise 51 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.
3. Test a value from each interval to see if it satisfies the original inequality.

Step 1

We will start by solving the related equation. 11 = 4x^2+7x ⇔ - 4x^2+( - 7)x+ 11=0We see above that a= - 4, b= - 7, and c= 11. Let's substitute these values into the Quadratic Formula to solve the equation.

Now we can calculate the first root using the positive sign and the second root using the negative sign.

Step 2

The solutions of the related equation are - 2.75 and 1. Let's plot them on a number line. Since the original is not a strict inequality, the points will be closed.

Step 3

Finally, we must test a value from each interval to see if it satisfies the original inequality. Let's choose a value from the first interval, x ≤ - 2.75. We will choose x=- 4.

Since x=- 4 produced a true statement, the interval x ≤ - 2.75 is a part of the solution. Similarly, we can test the other two intervals.

We can now write the solution set and show it on a number line. { x| x ≤ - 2.75 or x≥ 1 } or (- ∞, - 2.75] ⋃ [1, ∞)