Single Quotient

**blueridge** · June 10th 2007, 04:10 AM

Write each expression as a single quotient.

What does the author mean by single quotient?

(1) x/(x - 2) + x/(x^2 - 4)

(2) x/(x - 2) + x/(x^2 - 4) < 0

**topsquark** · June 10th 2007, 04:53 AM

Originally Posted by blueridge

Write each expression as a single quotient.

What does the author mean by single quotient?

(1) x/(x - 2) + x/(x^2 - 4)

(2) x/(x - 2) + x/(x^2 - 4) < 0

They are asking you to add the two fractions.

$x^2 - 4 = (x + 2)(x - 2)$

So the first problem says:
$\frac{x}{x - 2} + \frac{x}{(x + 2)(x - 2)}$

Can you take it from there?

-Dan

**blueridge** · June 10th 2007, 05:45 AM

Yes, but what what must I do to solve question 2 considering that it has a less than symbol?

**topsquark** · June 10th 2007, 08:06 AM

Originally Posted by blueridge

x/(x - 2) + x/(x^2 - 4) < 0

$\frac{x}{x - 2} + \frac{x}{x^2 - 4} < 0$

$\frac{x(x + 2) + x}{(x + 2)(x - 2)} < 0$

$\frac{x^2 + 3x}{(x + 2)(x - 2)} < 0$

$\frac{x(x + 3)}{(x + 2)(x - 2)} < 0$

Critical numbers are numbers where either the numerator or denominator is 0. In this case the critical numbers are -3, -2, 0, and 2.

So break up the real number line into 5 intervals and test the inequality on each interval:
$( -\infty, -3) \to \frac{x(x + 3)}{(x + 2)(x - 2)} > 0$ (No!)
$( -3, -2) \to \frac{x(x + 3)}{(x + 2)(x - 2)} < 0$ (Check!)
$( -2, 0) \to \frac{x(x + 3)}{(x + 2)(x - 2)} > 0$ (No!)
$( 0, 2) \to \frac{x(x + 3)}{(x + 2)(x - 2)} < 0$ (Check!)
$( 2, \infty ) \to \frac{x(x + 3)}{(x + 2)(x - 2)} > 0$ (No!)

So our solution set is $(-3, -2) \cup (0, 2)$ .

You can verify this on the graph attached below.

-Dan

**blueridge** · June 10th 2007, 11:13 AM

Is x/(x - 2) + x/(x^2 - 4) < 0 called a quadratic inequality?

What about if the symbol is the greater than (>) applied to the same question?

What about if the symbol is "greater than or equal to" or if the symbol is "less than or equal to"?

What would I do in those particular cases?

**topsquark** · June 10th 2007, 11:17 AM

Originally Posted by blueridge

Is x/(x - 2) + x/(x^2 - 4) < 0 called a quadratic inequality?

What about if the symbol is the greater than (>) applied to the same question?

What about if the symbol is "greater than or equal to" or if the symbol is "less than or equal to"?

What would I do in those particular cases?

I don't know about the name, but quadratic inequality sounds good.

As for what you do in the other cases, consider that there is an automatic restriction that the denominator not be zero, but depending on the type of inequality critical points that make the numerator zero may be allowed.

-Dan

**blueridge** · June 10th 2007, 12:28 PM

So, basically I approach the same question in terms of other inequality symbols in like manner, right?

The main idea is to select points from various intervals, right?

**Jhevon** · June 10th 2007, 12:31 PM

Originally Posted by blueridge

So, basically I approach the same question in terms of other inequality symbols in like manner, right?

The main idea is to select points from various intervals, right?

yes. same approach, just select the appropriate intervals

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