Algebra Examples

$x + 3 < \frac{1}{2} (4 x - 12) < 20$

Step 1

Solve $x + 3 < \frac{1}{2} \cdot (4 x - 12)$ for $x$ .

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Step 1.1

Simplify $\frac{1}{2} \cdot (4 x - 12)$ .

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Step 1.1.1

Rewrite.

$x + 3 < 0 + 0 + \frac{1}{2} \cdot (4 x - 12)$

Step 1.1.2

Simplify by adding zeros.

$x + 3 < \frac{1}{2} \cdot (4 x - 12)$

Step 1.1.3

Apply the distributive property.

$x + 3 < \frac{1}{2} (4 x) + \frac{1}{2} \cdot - 12$

Step 1.1.4

Cancel the common factor of $2$ .

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Step 1.1.4.1

Factor $2$ out of $4 x$ .

$x + 3 < \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12$

Step 1.1.4.2

Cancel the common factor.

$x + 3 < \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12$

Step 1.1.4.3

Rewrite the expression.

$x + 3 < 2 x + \frac{1}{2} \cdot - 12$

Step 1.1.5

Cancel the common factor of $2$ .

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Step 1.1.5.1

Factor $2$ out of $- 12$ .

$x + 3 < 2 x + \frac{1}{2} \cdot (2 (- 6))$

Step 1.1.5.2

Cancel the common factor.

$x + 3 < 2 x + \frac{1}{2} \cdot (2 \cdot - 6)$

Step 1.1.5.3

Rewrite the expression.

$x + 3 < 2 x - 6$

Step 1.2

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

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Step 1.2.1

Subtract $2 x$ from both sides of the inequality.

$x + 3 - 2 x < - 6$

Step 1.2.2

Subtract $2 x$ from $x$ .

$- x + 3 < - 6$

Step 1.3

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

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Step 1.3.1

Subtract $3$ from both sides of the inequality.

$- x < - 6 - 3$

Step 1.3.2

Subtract $3$ from $- 6$ .

$- x < - 9$

Step 1.4

Divide each term in $- x < - 9$ by $- 1$ and simplify.

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Step 1.4.1

Divide each term in $- x < - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 9}{- 1}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{- 9}{- 1}$

Step 1.4.2.2

Divide $x$ by $1$ .

$x > \frac{- 9}{- 1}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $- 9$ by $- 1$ .

$x > 9$

Step 2

Solve $\frac{1}{2} \cdot (4 x - 12) < 20$ for $x$ .

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Step 2.1

Simplify $\frac{1}{2} \cdot (4 x - 12)$ .

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Step 2.1.1

Apply the distributive property.

$\frac{1}{2} (4 x) + \frac{1}{2} \cdot - 12 < 20$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Factor $2$ out of $4 x$ .

$\frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12 < 20$

Step 2.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12 < 20$

Step 2.1.2.3

Rewrite the expression.

$2 x + \frac{1}{2} \cdot - 12 < 20$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Factor $2$ out of $- 12$ .

$2 x + \frac{1}{2} \cdot (2 (- 6)) < 20$

Step 2.1.3.2

Cancel the common factor.

$2 x + \frac{1}{2} \cdot (2 \cdot - 6) < 20$

Step 2.1.3.3

Rewrite the expression.

$2 x - 6 < 20$

Step 2.2

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

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Step 2.2.1

Add $6$ to both sides of the inequality.

$2 x < 20 + 6$

Step 2.2.2

Add $20$ and $6$ .

$2 x < 26$

Step 2.3

Divide each term in $2 x < 26$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x < 26$ by $2$ .

$\frac{2 x}{2} < \frac{26}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{26}{2}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{26}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $26$ by $2$ .

$x < 13$

Step 3

Find the intersection of $x > 9$ and $x < 13$ .

$9 < x < 13$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$9 < x < 13$

Interval Notation:

$(9, 13)$

Step 5