Algebra Examples

$- 2 > 5 + \frac{1}{3} (3 x + \frac{3}{4}) - \frac{3}{4} x$

Step 1

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

$5 + \frac{1}{3} \cdot (3 x + \frac{3}{4}) - \frac{3}{4} x < - 2$

Step 2

Simplify $5 + \frac{1}{3} \cdot (3 x + \frac{3}{4}) - \frac{3}{4} x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$5 + \frac{1}{3} (3 x) + \frac{1}{3} \cdot \frac{3}{4} - \frac{3}{4} x < - 2$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $3 x$ .

$5 + \frac{1}{3} (3 (x)) + \frac{1}{3} \cdot \frac{3}{4} - \frac{3}{4} x < - 2$

Step 2.1.2.2

Cancel the common factor.

$5 + \frac{1}{3} (3 x) + \frac{1}{3} \cdot \frac{3}{4} - \frac{3}{4} x < - 2$

Step 2.1.2.3

Rewrite the expression.

$5 + x + \frac{1}{3} \cdot \frac{3}{4} - \frac{3}{4} x < - 2$

Step 2.1.3

Cancel the common factor of $3$ .

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Step 2.1.3.1

Cancel the common factor.

$5 + x + \frac{1}{3} \cdot \frac{3}{4} - \frac{3}{4} x < - 2$

Step 2.1.3.2

Rewrite the expression.

$5 + x + \frac{1}{4} - \frac{3}{4} x < - 2$

Step 2.1.4

Combine $x$ and $\frac{3}{4}$ .

$5 + x + \frac{1}{4} - \frac{x \cdot 3}{4} < - 2$

Step 2.1.5

Move $3$ to the left of $x$ .

$5 + x + \frac{1}{4} - \frac{3 x}{4} < - 2$

Step 2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x + 5 \cdot \frac{4}{4} + \frac{1}{4} - \frac{3 x}{4} < - 2$

Step 2.3

Combine $5$ and $\frac{4}{4}$ .

$x + \frac{5 \cdot 4}{4} + \frac{1}{4} - \frac{3 x}{4} < - 2$

Step 2.4

Combine the numerators over the common denominator.

$x + \frac{5 \cdot 4 + 1}{4} - \frac{3 x}{4} < - 2$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $5$ by $4$ .

$x + \frac{20 + 1}{4} - \frac{3 x}{4} < - 2$

Step 2.5.2

Add $20$ and $1$ .

$x + \frac{21}{4} - \frac{3 x}{4} < - 2$

Step 2.6

To write $x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x \cdot \frac{4}{4} - \frac{3 x}{4} + \frac{21}{4} < - 2$

Step 2.7

Simplify terms.

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Step 2.7.1

Combine $x$ and $\frac{4}{4}$ .

$\frac{x \cdot 4}{4} - \frac{3 x}{4} + \frac{21}{4} < - 2$

Step 2.7.2

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - 3 x}{4} + \frac{21}{4} < - 2$

Step 2.7.3

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - 3 x + 21}{4} < - 2$

Step 2.8

Move $4$ to the left of $x$ .

$\frac{4 x - 3 x + 21}{4} < - 2$

Step 2.9

Subtract $3 x$ from $4 x$ .

$\frac{x + 21}{4} < - 2$

Step 3

Multiply both sides by $4$ .

$\frac{x + 21}{4} \cdot 4 < - 2 \cdot 4$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $4$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{x + 21}{4} \cdot 4 < - 2 \cdot 4$

Step 4.1.1.2

Rewrite the expression.

$x + 21 < - 2 \cdot 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 2$ by $4$ .

$x + 21 < - 8$

Step 5

Move all terms not containing $x$ to the right side of the inequality.

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Step 5.1

Subtract $21$ from both sides of the inequality.

$x < - 8 - 21$

Step 5.2

Subtract $21$ from $- 8$ .

$x < - 29$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - 29$

Interval Notation:

$(- \infty, - 29)$