MHB How Do You Solve This Quadratic Inequality: x - [10/(x - 1)] ≥ 4?

[mathdad]

Section 2.6
Question 68Solve the quadratic inequality.

x - [10/(x - 1)] ≥ 4

I begin by subtracting 4 from both sides and then simplify the left hand side, right?

 

[MarkFL]

Section 2.6
Question 68Solve the quadratic inequality.

x - [10/(x - 1)] ≥ 4

I begin by subtracting 4 from both sides and then simplify the left hand side, right?

Yes, you should wind up with:

$$\frac{(x+1)(x-6)}{x-1}\ge0$$

And then observing we have a weak inequality, and that $x=1$ must be excluded (division by zero), testing 1 interval, and allowing the others to alternate signs, we wind up with:

$$[-1,1)\,\cup\,[6,\infty)$$

 

[mathdad]

I will show my work tomorrow.

 

[mathdad]

(x + 1)(x - 6)/(x - 1) ≥ 0

We exclude x = 1 because it creates (expression/0) but include it on the number line.

<-------(-1)----------(1)---------(6)--------->

If we let x = -1 and 6, the inequality becomes 0 ≥ 0, which is true. So, we include our critical values of x.

For (-infinity, -1), let x = -2. False statement.

For (-1, 1), let x = 0. True statement.

For (1, 6), let x = 2. False statement.

For (6, infinity), let x = 7. True statement.

Solution:

[-1, 1) U [6, infinity)