What is -2<= (1/(x+4)-1/4)/x <= x+3 ?

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-2<= (1/(x+4)-1/4)/x <= x+3

- 2 \leq \frac{\frac{1}{x + 4} - \frac{1}{4}}{x} \leq x + 3

- \frac{31}{8} \leq x < 0 or x > 0

Interval Notation

[- \frac{31}{8}, 0) \cup (0, \infty)

Decimal Notation

- 3.875 \leq x < 0 or x > 0

Solution steps

- 2 \leq \frac{\frac{1}{x + 4} - \frac{1}{4}}{x} \leq x + 3

a \leq u \leq b

then

a \leq u and u \leq b

- 2 \leq \frac{\frac{1}{x + 4} - \frac{1}{4}}{x} and \frac{\frac{1}{x + 4} - \frac{1}{4}}{x} \leq x + 3

- 2 \leq \frac{\frac{1}{x + 4} - \frac{1}{4}}{x} : x < - 4 or - \frac{31}{8} \leq x < 0 or x > 0

\frac{\frac{1}{x + 4} - \frac{1}{4}}{x} \leq x + 3 : x > - 4

Combine the intervals

(x < - 4 or - \frac{31}{8} \leq x < 0 or x > 0) and x > - 4

Merge Overlapping Intervals

- \frac{31}{8} \leq x < 0 or x > 0

x (3 x - 4) \leq 7

1 - e^{- 2}

9 + \frac{14 - 2}{3 ^{2} + 3} - 3

7 x - 8 = 3 x + 12

x (8 x - 7) = 15

What is -2<= (1/(x+4)-1/4)/x <= x+3 ?
The solution to -2<= (1/(x+4)-1/4)/x <= x+3 is -31/8 <= x<0\lor x>0