Finite Math Examples

$3 - \sqrt{3} + 1 > (3 \sqrt{3} + 2) x$

Step 1

Rewrite so $x$ is on the left side of the inequality.

$(3 \sqrt{3} + 2) x < 3 - \sqrt{3} + 1$

Step 2

Add $3$ and $1$ .

$(3 \sqrt{3} + 2) x < 4 - \sqrt{3}$

Step 3

Divide each term in $(3 \sqrt{3} + 2) x < 4 - \sqrt{3}$ by $3 \sqrt{3} + 2$ and simplify.

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Step 3.1

Divide each term in $(3 \sqrt{3} + 2) x < 4 - \sqrt{3}$ by $3 \sqrt{3} + 2$ .

$\frac{(3 \sqrt{3} + 2) x}{3 \sqrt{3} + 2} < \frac{4}{3 \sqrt{3} + 2} + \frac{- \sqrt{3}}{3 \sqrt{3} + 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3 \sqrt{3} + 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(3 \sqrt{3} + 2) x}{3 \sqrt{3} + 2} < \frac{4}{3 \sqrt{3} + 2} + \frac{- \sqrt{3}}{3 \sqrt{3} + 2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x < \frac{4}{3 \sqrt{3} + 2} + \frac{- \sqrt{3}}{3 \sqrt{3} + 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$x < \frac{4 - \sqrt{3}}{3 \sqrt{3} + 2}$

Step 3.3.2

Multiply $\frac{4 - \sqrt{3}}{3 \sqrt{3} + 2}$ by $\frac{3 \sqrt{3} - 2}{3 \sqrt{3} - 2}$ .

$x < \frac{4 - \sqrt{3}}{3 \sqrt{3} + 2} \cdot \frac{3 \sqrt{3} - 2}{3 \sqrt{3} - 2}$

Step 3.3.3

Combine fractions.

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Step 3.3.3.1

Multiply $\frac{4 - \sqrt{3}}{3 \sqrt{3} + 2}$ by $\frac{3 \sqrt{3} - 2}{3 \sqrt{3} - 2}$ .

$x < \frac{(4 - \sqrt{3}) (3 \sqrt{3} - 2)}{(3 \sqrt{3} + 2) (3 \sqrt{3} - 2)}$

Step 3.3.3.2

Expand the denominator using the FOIL method.

$x < \frac{(4 - \sqrt{3}) (3 \sqrt{3} - 2)}{9 {\sqrt{3}}^{2} - 6 \sqrt{3} + 6 \sqrt{3} - 4}$

Step 3.3.3.3

Simplify.

$x < \frac{(4 - \sqrt{3}) (3 \sqrt{3} - 2)}{23}$

Step 3.3.4

Expand $(4 - \sqrt{3}) (3 \sqrt{3} - 2)$ using the FOIL Method.

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Step 3.3.4.1

Apply the distributive property.

$x < \frac{4 (3 \sqrt{3} - 2) - \sqrt{3} (3 \sqrt{3} - 2)}{23}$

Step 3.3.4.2

Apply the distributive property.

$x < \frac{4 (3 \sqrt{3}) + 4 \cdot - 2 - \sqrt{3} (3 \sqrt{3} - 2)}{23}$

Step 3.3.4.3

Apply the distributive property.

$x < \frac{4 (3 \sqrt{3}) + 4 \cdot - 2 - \sqrt{3} (3 \sqrt{3}) - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5

Simplify and combine like terms.

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Step 3.3.5.1

Simplify each term.

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Step 3.3.5.1.1

Multiply $3$ by $4$ .

$x < \frac{12 \sqrt{3} + 4 \cdot - 2 - \sqrt{3} (3 \sqrt{3}) - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.2

Multiply $4$ by $- 2$ .

$x < \frac{12 \sqrt{3} - 8 - \sqrt{3} (3 \sqrt{3}) - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.3

Multiply $- \sqrt{3} (3 \sqrt{3})$ .

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Step 3.3.5.1.3.1

Multiply $3$ by $- 1$ .

$x < \frac{12 \sqrt{3} - 8 - 3 \sqrt{3} \sqrt{3} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x < \frac{12 \sqrt{3} - 8 - 3 ({\sqrt{3}}^{1} \sqrt{3}) - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$x < \frac{12 \sqrt{3} - 8 - 3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x < \frac{12 \sqrt{3} - 8 - 3 {\sqrt{3}}^{1 + 1} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.3.5

Add $1$ and $1$ .

$x < \frac{12 \sqrt{3} - 8 - 3 {\sqrt{3}}^{2} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.5.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x < \frac{12 \sqrt{3} - 8 - 3 {(3^{\frac{1}{2}})}^{2} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x < \frac{12 \sqrt{3} - 8 - 3 \cdot 3^{\frac{1}{2} \cdot 2} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x < \frac{12 \sqrt{3} - 8 - 3 \cdot 3^{\frac{2}{2}} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.5.1.4.4.1

Cancel the common factor.

$x < \frac{12 \sqrt{3} - 8 - 3 \cdot 3^{\frac{2}{2}} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4.4.2

Rewrite the expression.

$x < \frac{12 \sqrt{3} - 8 - 3 \cdot 3^{1} - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.4.5

Evaluate the exponent.

$x < \frac{12 \sqrt{3} - 8 - 3 \cdot 3 - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.5

Multiply $- 3$ by $3$ .

$x < \frac{12 \sqrt{3} - 8 - 9 - \sqrt{3} \cdot - 2}{23}$

Step 3.3.5.1.6

Multiply $- 2$ by $- 1$ .

$x < \frac{12 \sqrt{3} - 8 - 9 + 2 \sqrt{3}}{23}$

Step 3.3.5.2

Add $12 \sqrt{3}$ and $2 \sqrt{3}$ .

$x < \frac{- 8 - 9 + 14 \sqrt{3}}{23}$

Step 3.3.5.3

Subtract $9$ from $- 8$ .

$x < \frac{- 17 + 14 \sqrt{3}}{23}$

Step 3.3.6

Rewrite $- 17$ as $- 1 (17)$ .

$x < \frac{- 1 (17) + 14 \sqrt{3}}{23}$

Step 3.3.7

Factor $- 1$ out of $14 \sqrt{3}$ .

$x < \frac{- 1 (17) - (- 14 \sqrt{3})}{23}$

Step 3.3.8

Factor $- 1$ out of $- 1 (17) - (- 14 \sqrt{3})$ .

$x < \frac{- 1 (17 - 14 \sqrt{3})}{23}$

Step 3.3.9

Move the negative in front of the fraction.

$x < - \frac{17 - 14 \sqrt{3}}{23}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{17 - 14 \sqrt{3}}{23}$

Interval Notation:

$(- \infty, - \frac{17 - 14 \sqrt{3}}{23})$