Algebra Examples

$- 1 < | 2 x - 3 | < 7$

Step 1

Find the $x$ values that make $- 1 < | 2 x - 3 |$ true.

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Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$| 2 x - 3 | > - 1$

Step 1.2

Since $| 2 x - 3 |$ is always positive and $- 1$ is negative, $| 2 x - 3 |$ is always greater than $- 1$ so the inequality is always true.

All real numbers

Step 2

Find the $x$ values that make $| 2 x - 3 | < 7$ true.

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Step 2.1

Write $| 2 x - 3 | < 7$ as a piecewise.

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Step 2.1.1

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

$2 x - 3 \geq 0$

Step 2.1.2

Solve the inequality.

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Step 2.1.2.1

Add $3$ to both sides of the inequality.

$2 x \geq 3$

Step 2.1.2.2

Divide each term in $2 x \geq 3$ by $2$ and simplify.

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Step 2.1.2.2.1

Divide each term in $2 x \geq 3$ by $2$ .

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

Step 2.1.2.2.2

Simplify the left side.

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Step 2.1.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.1.3

In the piece where $2 x - 3$ is non-negative, remove the absolute value.

$2 x - 3 < 7$

Step 2.1.4

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

$2 x - 3 < 0$

Step 2.1.5

Solve the inequality.

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Step 2.1.5.1

Add $3$ to both sides of the inequality.

$2 x < 3$

Step 2.1.5.2

Divide each term in $2 x < 3$ by $2$ and simplify.

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Step 2.1.5.2.1

Divide each term in $2 x < 3$ by $2$ .

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

Step 2.1.5.2.2

Simplify the left side.

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Step 2.1.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.5.2.2.1.1

Cancel the common factor.

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

Step 2.1.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.1.6

In the piece where $2 x - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 3) < 7$

Step 2.1.7

Write as a piecewise.

${\begin{cases} 2 x - 3 < 7 & x \geq \frac{3}{2} \\ - (2 x - 3) < 7 & x < \frac{3}{2} \end{cases}$

Step 2.1.8

Simplify $- (2 x - 3) < 7$ .

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Step 2.1.8.1

Apply the distributive property.

${\begin{cases} 2 x - 3 < 7 & x \geq \frac{3}{2} \\ - (2 x) - - 3 < 7 & x < \frac{3}{2} \end{cases}$

Step 2.1.8.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 3 < 7 & x \geq \frac{3}{2} \\ - 2 x - - 3 < 7 & x < \frac{3}{2} \end{cases}$

Step 2.1.8.3

Multiply $- 1$ by $- 3$ .

${\begin{cases} 2 x - 3 < 7 & x \geq \frac{3}{2} \\ - 2 x + 3 < 7 & x < \frac{3}{2} \end{cases}$

Step 2.2

Solve $2 x - 3 < 7$ when $x \geq \frac{3}{2}$ .

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Step 2.2.1

Solve $2 x - 3 < 7$ for $x$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Add $3$ to both sides of the inequality.

$2 x < 7 + 3$

Step 2.2.1.1.2

Add $7$ and $3$ .

$2 x < 10$

Step 2.2.1.2

Divide each term in $2 x < 10$ by $2$ and simplify.

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Step 2.2.1.2.1

Divide each term in $2 x < 10$ by $2$ .

$\frac{2 x}{2} < \frac{10}{2}$

Step 2.2.1.2.2

Simplify the left side.

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Step 2.2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{10}{2}$

Step 2.2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{10}{2}$

Step 2.2.1.2.3

Simplify the right side.

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Step 2.2.1.2.3.1

Divide $10$ by $2$ .

$x < 5$

Step 2.2.2

Find the intersection of $x < 5$ and $x \geq \frac{3}{2}$ .

$\frac{3}{2} \leq x < 5$

Step 2.3

Solve $- 2 x + 3 < 7$ when $x < \frac{3}{2}$ .

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Step 2.3.1

Solve $- 2 x + 3 < 7$ for $x$ .

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Subtract $3$ from both sides of the inequality.

$- 2 x < 7 - 3$

Step 2.3.1.1.2

Subtract $3$ from $7$ .

$- 2 x < 4$

Step 2.3.1.2

Divide each term in $- 2 x < 4$ by $- 2$ and simplify.

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Step 2.3.1.2.1

Divide each term in $- 2 x < 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} > \frac{4}{- 2}$

Step 2.3.1.2.2

Simplify the left side.

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Step 2.3.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} > \frac{4}{- 2}$

Step 2.3.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{4}{- 2}$

Step 2.3.1.2.3

Simplify the right side.

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Step 2.3.1.2.3.1

Divide $4$ by $- 2$ .

$x > - 2$

Step 2.3.2

Find the intersection of $x > - 2$ and $x < \frac{3}{2}$ .

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

Step 2.4

Find the union of the solutions.

$- 2 < x < 5$

Step 3

The solution is the intersection of the intervals.

All real numbers

$- 2 < x < 5$

Step 4

Find the intersection.

$- 2 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 5$

Interval Notation:

$(- 2, 5)$

Step 6