What is 1/(x+3)-1/(x+1)<2 ?

The solution to 1/(x+3)-1/(x+1) -1

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1/(x+3)-1/(x+1)<2

\frac{1}{x + 3} - \frac{1}{x + 1} < 2

x < - 3 or x > - 1

Interval Notation

(- \infty, - 3) \cup (- 1, \infty)

Solution steps

\frac{1}{x + 3} - \frac{1}{x + 1} < 2

Rewrite in standard form

\frac{- 2 x ^{2} - 8 x - 8}{( x + 3 ) ( x + 1 )} < 0

Factor

\frac{- 2 x ^{2} - 8 x - 8}{( x + 3 ) ( x + 1 )} : \frac{- 2 ( x + 2 ) ^{2}}{( x + 3 ) ( x + 1 )}

\frac{- 2 ( x + 2 ) ^{2}}{( x + 3 ) ( x + 1 )} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 2 ( x + 2 ) ^{2} ) ( - 1 )}{( x + 3 ) ( x + 1 )} > 0 \cdot (- 1)

Simplify

\frac{2 ( x + 2 ) ^{2}}{( x + 3 ) ( x + 1 )} > 0

Divide both sides by

2

\frac{\frac{2 ( x + 2 ) ^{2}}{( x + 3 ) ( x + 1 )}}{2} > \frac{0}{2}

Simplify

\frac{( x + 2 ) ^{2}}{( x + 3 ) ( x + 1 )} > 0

Identify the intervals

x < - 3 or x > - 1

0 - 3

∣ 2 x + 3 ∣ \geq 19

x^{2} + (5 x + 19)^{2} = (5 x + 20)^{2}

2 x^{3} - 9 x^{2} + 14 x - 5

i^{34}