Finite Math Examples

$| 2 x - 4 | < | x - 1 |$

Step 1

Replace $<$ with $=$ in $| 2 x - 4 | < | x - 1 |$ .

$| 2 x - 4 | = | x - 1 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$2 x - 4 = x - 1$

$2 x - 4 = - (x - 1)$

$- (2 x - 4) = x - 1$

$- (2 x - 4) = - (x - 1)$

Step 3

After simplifying, there are only two unique equations to be solved.

$2 x - 4 = x - 1$

$2 x - 4 = - (x - 1)$

Step 4

Solve $2 x - 4 = x - 1$ for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $x$ from both sides of the equation.

$2 x - 4 - x = - 1$

Step 4.1.2

Subtract $x$ from $2 x$ .

$x - 4 = - 1$

Step 4.2

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

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Step 4.2.1

Add $4$ to both sides of the equation.

$x = - 1 + 4$

Step 4.2.2

Add $- 1$ and $4$ .

$x = 3$

Step 5

Solve $2 x - 4 = - (x - 1)$ for $x$ .

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Step 5.1

Simplify $- (x - 1)$ .

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Step 5.1.1

Rewrite.

$2 x - 4 = 0 + 0 - (x - 1)$

Step 5.1.2

Simplify by adding zeros.

$2 x - 4 = - (x - 1)$

Step 5.1.3

Apply the distributive property.

$2 x - 4 = - x - - 1$

Step 5.1.4

Multiply $- 1$ by $- 1$ .

$2 x - 4 = - x + 1$

Step 5.2

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

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Step 5.2.1

Add $x$ to both sides of the equation.

$2 x - 4 + x = 1$

Step 5.2.2

Add $2 x$ and $x$ .

$3 x - 4 = 1$

Step 5.3

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

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Step 5.3.1

Add $4$ to both sides of the equation.

$3 x = 1 + 4$

Step 5.3.2

Add $1$ and $4$ .

$3 x = 5$

Step 5.4

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 5.4.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 6

List all of the solutions.

$x = 3, \frac{5}{3}$

Step 7

Use each root to create test intervals.

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

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

$x > 3$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.1

Test a value on the interval $x < \frac{5}{3}$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < \frac{5}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 8.1.2

Replace $x$ with $0$ in the original inequality.

$| 2 (0) - 4 | < | (0) - 1 |$

Step 8.1.3

The left side $4$ is not less than the right side $1$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $\frac{5}{3} < x < 3$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $\frac{5}{3} < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 8.2.2

Replace $x$ with $2$ in the original inequality.

$| 2 (2) - 4 | < | (2) - 1 |$

Step 8.2.3

The left side $0$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 8.3.2

Replace $x$ with $6$ in the original inequality.

$| 2 (6) - 4 | < | (6) - 1 |$

Step 8.3.3

The left side $8$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 8.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{5}{3}$ False

$\frac{5}{3} < x < 3$ True

$x > 3$ False

$x < \frac{5}{3}$ False

$\frac{5}{3} < x < 3$ True

$x > 3$ False

Step 9

The solution consists of all of the true intervals.

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

Step 10

Convert the inequality to interval notation.

$(\frac{5}{3}, 3)$

Step 11