Algebra Examples

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

Step 1

Simplify $\frac{3}{4} \cdot (x + 4) - \frac{3}{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

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

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

Step 1.1.3

Cancel the common factor of $4$ .

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Step 1.1.3.1

Cancel the common factor.

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

Step 1.1.3.2

Rewrite the expression.

$\frac{3 x}{4} + 3 - \frac{3}{2} \geq 3$

Step 1.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $3$ by $2$ .

$\frac{3 x}{4} + \frac{6 - 3}{2} \geq 3$

Step 1.5.2

Subtract $3$ from $6$ .

$\frac{3 x}{4} + \frac{3}{2} \geq 3$

Step 2

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

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Step 2.1

Subtract $\frac{3}{2}$ from both sides of the inequality.

$\frac{3 x}{4} \geq 3 - \frac{3}{2}$

Step 2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{4} \geq 3 \cdot \frac{2}{2} - \frac{3}{2}$

Step 2.3

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 x}{4} \geq \frac{3 \cdot 2}{2} - \frac{3}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} \geq \frac{3 \cdot 2 - 3}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $2$ .

$\frac{3 x}{4} \geq \frac{6 - 3}{2}$

Step 2.5.2

Subtract $3$ from $6$ .

$\frac{3 x}{4} \geq \frac{3}{2}$

Step 3

Multiply both sides by $4$ .

$\frac{3 x}{4} \cdot 4 \geq \frac{3}{2} \cdot 4$

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.

$\frac{3 x}{4} \cdot 4 \geq \frac{3}{2} \cdot 4$

Step 4.1.1.2

Rewrite the expression.

$3 x \geq \frac{3}{2} \cdot 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{2} \cdot 4$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $4$ .

$3 x \geq \frac{3}{2} \cdot (2 (2))$

Step 4.2.1.1.2

Cancel the common factor.

$3 x \geq \frac{3}{2} \cdot (2 \cdot 2)$

Step 4.2.1.1.3

Rewrite the expression.

$3 x \geq 3 \cdot 2$

Step 4.2.1.2

Multiply $3$ by $2$ .

$3 x \geq 6$

Step 5

Divide each term in $3 x \geq 6$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x \geq 6$ by $3$ .

$\frac{3 x}{3} \geq \frac{6}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{6}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{6}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $6$ by $3$ .

$x \geq 2$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \geq 2$

Interval Notation:

$[2, \infty)$