Calculus Examples

$\sqrt[4]{1 + 2 x} - 0.1 < 1 - \frac{1}{3} x < \sqrt[4]{1 + 2 x} + 0.1$

Step 1

Combine $x$ and $\frac{1}{3}$ .

$\sqrt[4]{1 + 2 x} - 0.1 < 1 - \frac{x}{3} < \sqrt[4]{1 + 2 x} + 0.1$

Step 2

Move all terms not containing $x$ from the center section of the inequality.

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Step 2.1

Subtract $1$ from each section of the inequality because it does not contain the variable we are trying to solve for.

$\sqrt[4]{1 + 2 x} - 0.1 - 1 < - \frac{x}{3} < \sqrt[4]{1 + 2 x} + 0.1 - 1$

Step 2.2

Subtract $1$ from $- 0.1$ .

$\sqrt[4]{1 + 2 x} - 1.1 < - \frac{x}{3} < \sqrt[4]{1 + 2 x} + 0.1 - 1$

Step 2.3

Multiply each term in the inequality by $3$ .

$\sqrt[4]{1 + 2 x} \cdot 3 - 1.1 \cdot 3 < - \frac{x}{3} \cdot 3 < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.4

Simplify each term.

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Step 2.4.1

Move $3$ to the left of $\sqrt[4]{1 + 2 x}$ .

$3 \cdot \sqrt[4]{1 + 2 x} - 1.1 \cdot 3 < - \frac{x}{3} \cdot 3 < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.4.2

Multiply $- 1.1$ by $3$ .

$3 \sqrt[4]{1 + 2 x} - 3.3 < - \frac{x}{3} \cdot 3 < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.5

Cancel the common factor of $3$ .

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Step 2.5.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$3 \sqrt[4]{1 + 2 x} - 3.3 < \frac{- x}{3} \cdot 3 < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.5.2

Cancel the common factor.

$3 \sqrt[4]{1 + 2 x} - 3.3 < \frac{- x}{3} \cdot 3 < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.5.3

Rewrite the expression.

$3 \sqrt[4]{1 + 2 x} - 3.3 < - x < \sqrt[4]{1 + 2 x} \cdot 3 + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.6

Simplify each term.

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Step 2.6.1

Move $3$ to the left of $\sqrt[4]{1 + 2 x}$ .

$3 \sqrt[4]{1 + 2 x} - 3.3 < - x < 3 \cdot \sqrt[4]{1 + 2 x} + 0.1 \cdot 3 - 1 \cdot 3$

Step 2.6.2

Multiply $0.1$ by $3$ .

$3 \sqrt[4]{1 + 2 x} - 3.3 < - x < 3 \sqrt[4]{1 + 2 x} + 0.3 - 1 \cdot 3$

Step 2.6.3

Multiply $- 1$ by $3$ .

$3 \sqrt[4]{1 + 2 x} - 3.3 < - x < 3 \sqrt[4]{1 + 2 x} + 0.3 - 3$

Step 2.7

Subtract $3$ from $0.3$ .

$3 \sqrt[4]{1 + 2 x} - 3.3 < - x < 3 \sqrt[4]{1 + 2 x} - 2.7$

Step 3

Multiply each term in the inequality by $- 1$ .

$\frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 3.3}{- 1} > \frac{- x}{- 1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 4

Simplify each term.

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Step 4.1

Move the negative one from the denominator of $\frac{3 \sqrt[4]{1 + 2 x}}{- 1}$ .

$- 1 \cdot (3 \sqrt[4]{1 + 2 x}) + \frac{- 3.3}{- 1} > \frac{- x}{- 1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 4.2

Rewrite $- 1 \cdot (3 \sqrt[4]{1 + 2 x})$ as $- (3 \sqrt[4]{1 + 2 x})$ .

$- (3 \sqrt[4]{1 + 2 x}) + \frac{- 3.3}{- 1} > \frac{- x}{- 1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 4.3

Multiply $3$ by $- 1$ .

$- 3 \sqrt[4]{1 + 2 x} + \frac{- 3.3}{- 1} > \frac{- x}{- 1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 4.4

Divide $- 3.3$ by $- 1$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > \frac{- x}{- 1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 5

Dividing two negative values results in a positive value.

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > \frac{x}{1} > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 6

Divide $x$ by $1$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > x > \frac{3 \sqrt[4]{1 + 2 x}}{- 1} + \frac{- 2.7}{- 1}$

Step 7

Simplify each term.

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Step 7.1

Move the negative one from the denominator of $\frac{3 \sqrt[4]{1 + 2 x}}{- 1}$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > x > - 1 \cdot (3 \sqrt[4]{1 + 2 x}) + \frac{- 2.7}{- 1}$

Step 7.2

Rewrite $- 1 \cdot (3 \sqrt[4]{1 + 2 x})$ as $- (3 \sqrt[4]{1 + 2 x})$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > x > - (3 \sqrt[4]{1 + 2 x}) + \frac{- 2.7}{- 1}$

Step 7.3

Multiply $3$ by $- 1$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > x > - 3 \sqrt[4]{1 + 2 x} + \frac{- 2.7}{- 1}$

Step 7.4

Divide $- 2.7$ by $- 1$ .

$- 3 \sqrt[4]{1 + 2 x} + 3.3 > x > - 3 \sqrt[4]{1 + 2 x} + 2.7$

Step 8

Convert the inequality to interval notation.

$(- 3 \sqrt[4]{1 + 2 x} + 3.3, - 3 \sqrt[4]{1 + 2 x} + 2.7)$

Step 9