Algebra Examples

$- \frac{3}{2} x < 6 - (2 x - 5)$

Step 1

Simplify $- \frac{3}{2} x$ .

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Step 1.1

Rewrite.

$0 + 0 - \frac{3}{2} x < 6 - (2 x - 5)$

Step 1.2

Simplify by adding zeros.

$- \frac{3}{2} x < 6 - (2 x - 5)$

Step 1.3

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

$- \frac{x \cdot 3}{2} < 6 - (2 x - 5)$

Step 1.4

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

$- \frac{3 x}{2} < 6 - (2 x - 5)$

$- \frac{3 x}{2} < 6 - (2 x - 5)$

Step 2

Simplify $6 - (2 x - 5)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{3 x}{2} < 6 - (2 x) - - 5$

Step 2.1.2

Multiply $2$ by $- 1$ .

$- \frac{3 x}{2} < 6 - 2 x - - 5$

Step 2.1.3

Multiply $- 1$ by $- 5$ .

$- \frac{3 x}{2} < 6 - 2 x + 5$

$- \frac{3 x}{2} < 6 - 2 x + 5$

Step 2.2

Add $6$ and $5$ .

$- \frac{3 x}{2} < - 2 x + 11$

$- \frac{3 x}{2} < - 2 x + 11$

Step 3

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

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Step 3.1

Add $2 x$ to both sides of the inequality.

$- \frac{3 x}{2} + 2 x < 11$

Step 3.2

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

$- \frac{3 x}{2} + 2 x \cdot \frac{2}{2} < 11$

Step 3.3

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

$- \frac{3 x}{2} + \frac{2 x \cdot 2}{2} < 11$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- 3 x + 2 x \cdot 2}{2} < 11$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $x$ out of $- 3 x + 2 x \cdot 2$ .

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Step 3.5.1.1

Factor $x$ out of $- 3 x$ .

$\frac{x \cdot - 3 + 2 x \cdot 2}{2} < 11$

Step 3.5.1.2

Factor $x$ out of $2 x \cdot 2$ .

$\frac{x \cdot - 3 + x (2 \cdot 2)}{2} < 11$

Step 3.5.1.3

Factor $x$ out of $x \cdot - 3 + x (2 \cdot 2)$ .

$\frac{x (- 3 + 2 \cdot 2)}{2} < 11$

$\frac{x (- 3 + 2 \cdot 2)}{2} < 11$

Step 3.5.2

Multiply $2$ by $2$ .

$\frac{x (- 3 + 4)}{2} < 11$

Step 3.5.3

Add $- 3$ and $4$ .

$\frac{x \cdot 1}{2} < 11$

$\frac{x \cdot 1}{2} < 11$

Step 3.6

Multiply $x$ by $1$ .

$\frac{x}{2} < 11$

$\frac{x}{2} < 11$

Step 4

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 < 11 \cdot 2$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 < 11 \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$x < 11 \cdot 2$

$x < 11 \cdot 2$

$x < 11 \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $11$ by $2$ .

$x < 22$

$x < 22$

$x < 22$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < 22$

Interval Notation:

$(- \infty, 22)$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$