Algebra Examples

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

Step 1

Simplify $\frac{1}{2} \cdot (x - 3)$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

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

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

Step 1.5

Combine $\frac{1}{2}$ and $- 3$ .

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

Step 1.6

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Subtract $x$ from $x \cdot 2$ .

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Step 2.5.1

Reorder $x$ and $2$ .

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

Step 2.5.2

Subtract $x$ from $2 \cdot x$ .

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

Step 3

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

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Step 3.1

Subtract $4$ from both sides of the inequality.

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

Step 3.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3

Combine $- 4$ and $\frac{2}{2}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 4$ by $2$ .

$\frac{x}{2} \geq \frac{- 3 - 8}{2}$

Step 3.5.2

Subtract $8$ from $- 3$ .

$\frac{x}{2} \geq \frac{- 11}{2}$

Step 3.6

Move the negative in front of the fraction.

$\frac{x}{2} \geq - \frac{11}{2}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x \geq - 11$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \geq - 11$

Interval Notation:

$[- 11, \infty)$