Algebra Examples

$\frac{1}{3} (4 x + 3) \geq \frac{2}{3} x + 2$

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{3} \cdot (4 x + 3) \geq \frac{2}{3} x + 2$

Step 1.2

Simplify by adding zeros.

$\frac{1}{3} \cdot (4 x + 3) \geq \frac{2}{3} x + 2$

Step 1.3

Apply the distributive property.

$\frac{1}{3} (4 x) + \frac{1}{3} \cdot 3 \geq \frac{2}{3} x + 2$

Step 1.4

Multiply $\frac{1}{3} (4 x)$ .

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Step 1.4.1

Combine $4$ and $\frac{1}{3}$ .

$\frac{4}{3} x + \frac{1}{3} \cdot 3 \geq \frac{2}{3} x + 2$

Step 1.4.2

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

$\frac{4 x}{3} + \frac{1}{3} \cdot 3 \geq \frac{2}{3} x + 2$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Cancel the common factor.

$\frac{4 x}{3} + \frac{1}{3} \cdot 3 \geq \frac{2}{3} x + 2$

Step 1.5.2

Rewrite the expression.

$\frac{4 x}{3} + 1 \geq \frac{2}{3} x + 2$

Step 2

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

$\frac{4 x}{3} + 1 \geq \frac{2 x}{3} + 2$

Step 3

Move all terms containing $x$ to the left side of the inequality.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $2 x$ from $4 x$ .

$1 + \frac{2 x}{3} \geq 2$

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the inequality.

$\frac{2 x}{3} \geq 2 - 1$

Step 4.2

Subtract $1$ from $2$ .

$\frac{2 x}{3} \geq 1$

Step 5

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 \geq 1 \cdot 3$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 \geq 1 \cdot 3$

Step 6.1.1.2

Rewrite the expression.

$2 x \geq 1 \cdot 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $3$ by $1$ .

$2 x \geq 3$

Step 7

Divide each term in $2 x \geq 3$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 x \geq 3$ by $2$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$[\frac{3}{2}, \infty)$