Precalculus Examples

$| \frac{1}{2} x - 1 | + \frac{3}{2} x < 10$

Step 1

Write $| \frac{1}{2} x - 1 | + \frac{3}{2} x < 10$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{1}{2} x - 1 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

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

$\frac{x}{2} - 1 \geq 0$

Step 1.2.2

Add $1$ to both sides of the inequality.

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

Step 1.2.3

Multiply both sides by $2$ .

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Cancel the common factor of $2$ .

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Step 1.2.4.1.1.1

Cancel the common factor.

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

Step 1.2.4.1.1.2

Rewrite the expression.

$x \geq 1 \cdot 2$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $2$ by $1$ .

$x \geq 2$

Step 1.3

In the piece where $\frac{1}{2} x - 1$ is non-negative, remove the absolute value.

$\frac{1}{2} x - 1 + \frac{3}{2} x < 10$

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{1}{2} x - 1 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

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

$\frac{x}{2} - 1 < 0$

Step 1.5.2

Add $1$ to both sides of the inequality.

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

Step 1.5.3

Multiply both sides by $2$ .

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

Step 1.5.4

Simplify.

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Step 1.5.4.1

Simplify the left side.

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Step 1.5.4.1.1

Cancel the common factor of $2$ .

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Step 1.5.4.1.1.1

Cancel the common factor.

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

Step 1.5.4.1.1.2

Rewrite the expression.

$x < 1 \cdot 2$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Multiply $2$ by $1$ .

$x < 2$

Step 1.6

In the piece where $\frac{1}{2} x - 1$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{1}{2} x - 1) + \frac{3}{2} x < 10$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{1}{2} x - 1 + \frac{3}{2} x < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8

Simplify $\frac{1}{2} x - 1 + \frac{3}{2} x < 10$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

${\begin{cases} \frac{x}{2} - 1 + \frac{3}{2} x < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.1.2

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

${\begin{cases} \frac{x}{2} - 1 + \frac{3 x}{2} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.2

Combine the numerators over the common denominator.

${\begin{cases} - 1 + \frac{x + 3 x}{2} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.3

Add $x$ and $3 x$ .

${\begin{cases} - 1 + \frac{4 x}{2} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.4

Cancel the common factor of $4$ and $2$ .

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Step 1.8.4.1

Factor $2$ out of $4 x$ .

${\begin{cases} - 1 + \frac{2 (2 x)}{2} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.4.2

Cancel the common factors.

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Step 1.8.4.2.1

Factor $2$ out of $2$ .

${\begin{cases} - 1 + \frac{2 (2 x)}{2 (1)} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.4.2.2

Cancel the common factor.

${\begin{cases} - 1 + \frac{2 (2 x)}{2 \cdot 1} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.4.2.3

Rewrite the expression.

${\begin{cases} - 1 + \frac{2 x}{1} < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.8.4.2.4

Divide $2 x$ by $1$ .

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ - (\frac{1}{2} x - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.9

Simplify $- (\frac{1}{2} x - 1) + \frac{3}{2} x < 10$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ - (\frac{x}{2} - 1) + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.9.1.2

Apply the distributive property.

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ - \frac{x}{2} - - 1 + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 1$ .

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ - \frac{x}{2} + 1 + \frac{3}{2} x < 10 & x < 2 \end{cases}$

Step 1.9.1.4

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

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ - \frac{x}{2} + 1 + \frac{3 x}{2} < 10 & x < 2 \end{cases}$

Step 1.9.2

Combine the numerators over the common denominator.

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ 1 + \frac{- x + 3 x}{2} < 10 & x < 2 \end{cases}$

Step 1.9.3

Add $- x$ and $3 x$ .

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ 1 + \frac{2 x}{2} < 10 & x < 2 \end{cases}$

Step 1.9.4

Cancel the common factor of $2$ .

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Step 1.9.4.1

Cancel the common factor.

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ 1 + \frac{2 x}{2} < 10 & x < 2 \end{cases}$

Step 1.9.4.2

Divide $x$ by $1$ .

${\begin{cases} - 1 + 2 x < 10 & x \geq 2 \\ 1 + x < 10 & x < 2 \end{cases}$

Step 2

Solve $- 1 + 2 x < 10$ when $x \geq 2$ .

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Step 2.1

Solve $- 1 + 2 x < 10$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Add $1$ to both sides of the inequality.

$2 x < 10 + 1$

Step 2.1.1.2

Add $10$ and $1$ .

$2 x < 11$

Step 2.1.2

Divide each term in $2 x < 11$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x < 11$ by $2$ .

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

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2

Find the intersection of $x < \frac{11}{2}$ and $x \geq 2$ .

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

Step 3

Solve $1 + x < 10$ when $x < 2$ .

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Step 3.1

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

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Step 3.1.1

Subtract $1$ from both sides of the inequality.

$x < 10 - 1$

Step 3.1.2

Subtract $1$ from $10$ .

$x < 9$

Step 3.2

Find the intersection of $x < 9$ and $x < 2$ .

$x < 2$

Step 4

Find the union of the solutions.

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

Step 5

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, \frac{11}{2})$

Step 6