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Rational inequalities

+2 votes

(x+1)/(x-2)<(x-1)/(x+2)?

asked Feb 8, 2013 in ALGEBRA 1 by chrisgirl Apprentice

4 Answers

+3 votes

(x + 1)/(x - 2) < (x - 1)/(x + 2)

Cross multiplication.

(x + 1)(x + 2)<(x - 1)(x - 2).

FOIL method: the product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms and the Last terms.

x2 + 2x + x + 2 < x2 - 2x - x + 2

x2 + 3x + 2 < x2 - 3x + 2

Subtract x2 & 2 from each side.

3x < - 3x

Add 3x to each side.

6x < 0

Divide each side by 6.

x < 0

Graph the solution set on a number line.

answered Feb 9, 2013 by richardson Scholar
0 votes

(x + 1)/(x − 2) < (x − 1)/(x + 2)
(x + 1)/(x − 2) − (x − 1)/(x + 2) < 0
((x + 1)(x + 2) − (x − 2)(x − 1))/((x − 2)(x + 2)) < 0
(x² + 3x + 2 − x² + 3x − 2)/((x − 2)(x + 2)) < 0
(6x)/((x − 2)(x + 2)) < 0

Roots of the numerator:
6x = 0 ⇒ x = 0. For x > 0, the numerator is positive. For x < 0, the numerator is negative. (crescent function)

Roots of the denominator (remember, it CAN'T be zero):
(x − 2)(x + 2) ≠ 0 ⇒ x ≠ 2, x ≠ −2. For |x| > 2, we have that the denominator is positive. Reciprocally, for |x| < 2, the denominator is negative.

So, our goal is to have both the numerator and the denominator with different signs (so it assumes a negative value, that is less than 0).

x > 2: +/+ = +, not what we want.
2 > x > 0: +/− = −
−2 < x < 0: −/− = +, not what we want.
x < −2: −/+ = −

So, the solution is, in set notation:
S = {x ∈ ℝ | x < −2 or 0 < x < 2}

Or, in interval notation:
S = (∞⁻, −2) U (0, 2)

Source: http://www.answers.yahoo.com/

answered Mar 26, 2013 by casacop Expert
0 votes

The rational inequality is image.

When solving a rational inequality, begin by writing the inequality in general form with the rational expression

on the left and zero on the right.

image

image

image

image

Now, the rational inequality is image.

Step - 1 :
 

State the exclude values, those are the values for which denominator is zero.

The exclude value of the inequality is 2 and - 2.

Step - 2 :

Solve the related equation image

6x = 0

x = 0.

Solution of related equation x = 0.

Step - 3 :

Draw the vertical lines at the exclude values and at the solution to separate the number line into intervals.

image

answered Jul 3, 2014 by lilly Expert
0 votes

Contd.......

Step - 4 :

Now test  sample values in each interval to determine whether values in the interval satisify the inequality.

Test interval x - value Inequality   Conclusion
(- ∞, -2) x = - 3 image True
(- 2, 0) x = - 1 image False
(0, 2) x = 1 image True
(2, ∞) x = 3 image False

 

Note that the original inequality contains a “ < ” symbol, We exlude it into set of solutions at x = - 2

image

Above statement is true.

x < - 2 is a solution of inequality.

The above conclude that the inequality is satisfied for all x - values in (- ∞, -2) and (0, 2).

This implies that the solution  of  the  inequalityimage is  the  interval (- ∞, - 2) and (0, 2) . as shown in Figure below. Note that the original inequality contains a “ < ” symbol. This means that the solution set does not contain the endpoints of the test interval is (- ∞, - 2) .

image

Solution of the inequality image is { x | x < - 2 or 0 < x < 2 }.
 

The interval notation form of inequality is (- ∞, - 2) U (0, 2).

answered Jul 3, 2014 by lilly Expert

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