Algebra Examples

$3 \frac{x}{2} + 3 | x - 2 | \leq 3$

Step 1

Write $3 (\frac{x}{2}) + 3 | x - 2 | \leq 3$ 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.

$x - 2 \geq 0$

Step 1.2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 1.3

In the piece where $x - 2$ is non-negative, remove the absolute value.

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

Step 1.4

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

$x - 2 < 0$

Step 1.5

Add $2$ to both sides of the inequality.

$x < 2$

Step 1.6

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

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

Step 1.7

Write as a piecewise.

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

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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

Step 1.8.1.2

Apply the distributive property.

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

Step 1.8.1.3

Multiply $3$ by $- 2$ .

${\begin{cases} \frac{3 x}{2} + 3 x - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.2

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

${\begin{cases} \frac{3 x}{2} + 3 x \cdot \frac{2}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.3

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

${\begin{cases} \frac{3 x}{2} + \frac{3 x \cdot 2}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.4

Combine the numerators over the common denominator.

${\begin{cases} \frac{3 x + 3 x \cdot 2}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.5

Simplify the numerator.

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Step 1.8.5.1

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

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Step 1.8.5.1.1

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

${\begin{cases} \frac{3 x (1) + 3 x \cdot 2}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.5.1.2

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

${\begin{cases} \frac{3 x (1) + 3 x (2)}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.5.1.3

Factor $3 x$ out of $3 x (1) + 3 x (2)$ .

${\begin{cases} \frac{3 x (1 + 2)}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.5.2

Add $1$ and $2$ .

${\begin{cases} \frac{3 x \cdot 3}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.8.5.3

Multiply $3$ by $3$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ 3 (\frac{x}{2}) + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.9

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

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} + 3 (- (x - 2)) \leq 3 & x < 2 \end{cases}$

Step 1.9.1.2

Apply the distributive property.

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} + 3 (- x - - 2) \leq 3 & x < 2 \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 2$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} + 3 (- x + 2) \leq 3 & x < 2 \end{cases}$

Step 1.9.1.4

Apply the distributive property.

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} + 3 (- x) + 3 \cdot 2 \leq 3 & x < 2 \end{cases}$

Step 1.9.1.5

Multiply $- 1$ by $3$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} - 3 x + 3 \cdot 2 \leq 3 & x < 2 \end{cases}$

Step 1.9.1.6

Multiply $3$ by $2$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} - 3 x + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.2

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

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} - 3 x \cdot \frac{2}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.3

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

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x}{2} + \frac{- 3 x \cdot 2}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.4

Combine the numerators over the common denominator.

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x - 3 x \cdot 2}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5

Simplify each term.

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Step 1.9.5.1

Simplify the numerator.

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Step 1.9.5.1.1

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

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Step 1.9.5.1.1.1

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

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x (1) - 3 x \cdot 2}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.1.1.2

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

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x (1) + 3 x (- 1 \cdot 2)}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.1.1.3

Factor $3 x$ out of $3 x (1) + 3 x (- 1 \cdot 2)$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x (1 - 1 \cdot 2)}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.1.2

Multiply $- 1$ by $2$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x (1 - 2)}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.1.3

Subtract $2$ from $1$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{3 x \cdot - 1}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.1.4

Multiply $- 1$ by $3$ .

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ \frac{- 3 x}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 1.9.5.2

Move the negative in front of the fraction.

${\begin{cases} \frac{9 x}{2} - 6 \leq 3 & x \geq 2 \\ - \frac{3 x}{2} + 6 \leq 3 & x < 2 \end{cases}$

Step 2

Solve $\frac{9 x}{2} - 6 \leq 3$ when $x \geq 2$ .

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Step 2.1

Solve $\frac{9 x}{2} - 6 \leq 3$ 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 $6$ to both sides of the inequality.

$\frac{9 x}{2} \leq 3 + 6$

Step 2.1.1.2

Add $3$ and $6$ .

$\frac{9 x}{2} \leq 9$

Step 2.1.2

Multiply both sides by $2$ .

$\frac{9 x}{2} \cdot 2 \leq 9 \cdot 2$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Simplify the left side.

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Step 2.1.3.1.1

Cancel the common factor of $2$ .

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Step 2.1.3.1.1.1

Cancel the common factor.

$\frac{9 x}{2} \cdot 2 \leq 9 \cdot 2$

Step 2.1.3.1.1.2

Rewrite the expression.

$9 x \leq 9 \cdot 2$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Multiply $9$ by $2$ .

$9 x \leq 18$

Step 2.1.4

Divide each term in $9 x \leq 18$ by $9$ and simplify.

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Step 2.1.4.1

Divide each term in $9 x \leq 18$ by $9$ .

$\frac{9 x}{9} \leq \frac{18}{9}$

Step 2.1.4.2

Simplify the left side.

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Step 2.1.4.2.1

Cancel the common factor of $9$ .

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Step 2.1.4.2.1.1

Cancel the common factor.

$\frac{9 x}{9} \leq \frac{18}{9}$

Step 2.1.4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{18}{9}$

Step 2.1.4.3

Simplify the right side.

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Step 2.1.4.3.1

Divide $18$ by $9$ .

$x \leq 2$

Step 2.2

Find the intersection of $x \leq 2$ and $x \geq 2$ .

$x = 2$

Step 3

Solve $- \frac{3 x}{2} + 6 \leq 3$ when $x < 2$ .

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Step 3.1

Solve $- \frac{3 x}{2} + 6 \leq 3$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $6$ from both sides of the inequality.

$- \frac{3 x}{2} \leq 3 - 6$

Step 3.1.1.2

Subtract $6$ from $3$ .

$- \frac{3 x}{2} \leq - 3$

Step 3.1.2

Divide each term in $- \frac{3 x}{2} \leq - 3$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- \frac{3 x}{2} \leq - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.2.2.2

Divide $\frac{3 x}{2}$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 3$ by $- 1$ .

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

Step 3.1.3

Multiply both sides by $2$ .

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

Step 3.1.4

Simplify.

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Step 3.1.4.1

Simplify the left side.

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Step 3.1.4.1.1

Cancel the common factor of $2$ .

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Step 3.1.4.1.1.1

Cancel the common factor.

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

Step 3.1.4.1.1.2

Rewrite the expression.

$3 x \geq 3 \cdot 2$

Step 3.1.4.2

Simplify the right side.

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Step 3.1.4.2.1

Multiply $3$ by $2$ .

$3 x \geq 6$

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

Simplify the left side.

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Step 3.1.5.2.1

Cancel the common factor of $3$ .

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Step 3.1.5.2.1.1

Cancel the common factor.

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

Step 3.1.5.2.1.2

Divide $x$ by $1$ .

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

Step 3.1.5.3

Simplify the right side.

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Step 3.1.5.3.1

Divide $6$ by $3$ .

$x \geq 2$

Step 3.2

Find the intersection of $x \geq 2$ and $x < 2$ .

No solution

Step 4

Find the union of the solutions.

$x = 2$

Step 5