Algebra Examples

$| (- 2 x + 9) (x - 1) | < 5 | x - 1 |$

Step 1

Replace $<$ with $=$ in $| (- 2 x + 9) (x - 1) | < 5 | x - 1 |$ .

$| (- 2 x + 9) (x - 1) | = 5 | x - 1 |$

Step 2

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

$(- 2 x + 9) (x - 1) = 5 (x - 1)$

$(- 2 x + 9) (x - 1) = - 5 (x - 1)$

$- (- 2 x + 9) (x - 1) = 5 (x - 1)$

$- (- 2 x + 9) (x - 1) = - 5 (x - 1)$

Step 3

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

$(- 2 x + 9) (x - 1) = 5 (x - 1)$

$(- 2 x + 9) (x - 1) = - 5 (x - 1)$

Step 4

Solve $(- 2 x + 9) (x - 1) = 5 (x - 1)$ for $x$ .

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Step 4.1

Simplify $(- 2 x + 9) (x - 1)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (- 2 x + 9) (x - 1) = 5 (x - 1)$

Step 4.1.2

Simplify by adding zeros.

$(- 2 x + 9) (x - 1) = 5 (x - 1)$

Step 4.1.3

Expand $(- 2 x + 9) (x - 1)$ using the FOIL Method.

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Step 4.1.3.1

Apply the distributive property.

$- 2 x (x - 1) + 9 (x - 1) = 5 (x - 1)$

Step 4.1.3.2

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 1 + 9 (x - 1) = 5 (x - 1)$

Step 4.1.3.3

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = 5 (x - 1)$

Step 4.1.4

Simplify and combine like terms.

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Step 4.1.4.1

Simplify each term.

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Step 4.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.4.1.1.1

Move $x$ .

$- 2 (x \cdot x) - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = 5 (x - 1)$

Step 4.1.4.1.1.2

Multiply $x$ by $x$ .

$- 2 x^{2} - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = 5 (x - 1)$

Step 4.1.4.1.2

Multiply $- 1$ by $- 2$ .

$- 2 x^{2} + 2 x + 9 x + 9 \cdot - 1 = 5 (x - 1)$

Step 4.1.4.1.3

Multiply $9$ by $- 1$ .

$- 2 x^{2} + 2 x + 9 x - 9 = 5 (x - 1)$

Step 4.1.4.2

Add $2 x$ and $9 x$ .

$- 2 x^{2} + 11 x - 9 = 5 (x - 1)$

Step 4.2

Simplify $5 (x - 1)$ .

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Step 4.2.1

Apply the distributive property.

$- 2 x^{2} + 11 x - 9 = 5 x + 5 \cdot - 1$

Step 4.2.2

Multiply $5$ by $- 1$ .

$- 2 x^{2} + 11 x - 9 = 5 x - 5$

Step 4.3

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

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Step 4.3.1

Subtract $5 x$ from both sides of the equation.

$- 2 x^{2} + 11 x - 9 - 5 x = - 5$

Step 4.3.2

Subtract $5 x$ from $11 x$ .

$- 2 x^{2} + 6 x - 9 = - 5$

Step 4.4

Add $5$ to both sides of the equation.

$- 2 x^{2} + 6 x - 9 + 5 = 0$

Step 4.5

Add $- 9$ and $5$ .

$- 2 x^{2} + 6 x - 4 = 0$

Step 4.6

Factor the left side of the equation.

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Step 4.6.1

Factor $- 2$ out of $- 2 x^{2} + 6 x - 4$ .

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Step 4.6.1.1

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} + 6 x - 4 = 0$

Step 4.6.1.2

Factor $- 2$ out of $6 x$ .

$- 2 x^{2} - 2 (- 3 x) - 4 = 0$

Step 4.6.1.3

Factor $- 2$ out of $- 4$ .

$- 2 x^{2} - 2 (- 3 x) - 2 \cdot 2 = 0$

Step 4.6.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 3 x)$ .

$- 2 (x^{2} - 3 x) - 2 \cdot 2 = 0$

Step 4.6.1.5

Factor $- 2$ out of $- 2 (x^{2} - 3 x) - 2 (2)$ .

$- 2 (x^{2} - 3 x + 2) = 0$

Step 4.6.2

Factor.

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Step 4.6.2.1

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 4.6.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 4.6.2.1.2

Write the factored form using these integers.

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

Step 4.6.2.2

Remove unnecessary parentheses.

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

Step 4.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x - 1 = 0$

Step 4.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.9

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.9.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.10

The final solution is all the values that make $- 2 (x - 2) (x - 1) = 0$ true.

$x = 2, 1$

Step 5

Solve $(- 2 x + 9) (x - 1) = - 5 (x - 1)$ for $x$ .

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Step 5.1

Simplify $(- 2 x + 9) (x - 1)$ .

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Step 5.1.1

Rewrite.

$0 + 0 + (- 2 x + 9) (x - 1) = - 5 (x - 1)$

Step 5.1.2

Simplify by adding zeros.

$(- 2 x + 9) (x - 1) = - 5 (x - 1)$

Step 5.1.3

Expand $(- 2 x + 9) (x - 1)$ using the FOIL Method.

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Step 5.1.3.1

Apply the distributive property.

$- 2 x (x - 1) + 9 (x - 1) = - 5 (x - 1)$

Step 5.1.3.2

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 1 + 9 (x - 1) = - 5 (x - 1)$

Step 5.1.3.3

Apply the distributive property.

$- 2 x \cdot x - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = - 5 (x - 1)$

Step 5.1.4

Simplify and combine like terms.

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Step 5.1.4.1

Simplify each term.

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Step 5.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.4.1.1.1

Move $x$ .

$- 2 (x \cdot x) - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = - 5 (x - 1)$

Step 5.1.4.1.1.2

Multiply $x$ by $x$ .

$- 2 x^{2} - 2 x \cdot - 1 + 9 x + 9 \cdot - 1 = - 5 (x - 1)$

Step 5.1.4.1.2

Multiply $- 1$ by $- 2$ .

$- 2 x^{2} + 2 x + 9 x + 9 \cdot - 1 = - 5 (x - 1)$

Step 5.1.4.1.3

Multiply $9$ by $- 1$ .

$- 2 x^{2} + 2 x + 9 x - 9 = - 5 (x - 1)$

Step 5.1.4.2

Add $2 x$ and $9 x$ .

$- 2 x^{2} + 11 x - 9 = - 5 (x - 1)$

Step 5.2

Simplify $- 5 (x - 1)$ .

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Step 5.2.1

Apply the distributive property.

$- 2 x^{2} + 11 x - 9 = - 5 x - 5 \cdot - 1$

Step 5.2.2

Multiply $- 5$ by $- 1$ .

$- 2 x^{2} + 11 x - 9 = - 5 x + 5$

Step 5.3

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

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Step 5.3.1

Add $5 x$ to both sides of the equation.

$- 2 x^{2} + 11 x - 9 + 5 x = 5$

Step 5.3.2

Add $11 x$ and $5 x$ .

$- 2 x^{2} + 16 x - 9 = 5$

Step 5.4

Subtract $5$ from both sides of the equation.

$- 2 x^{2} + 16 x - 9 - 5 = 0$

Step 5.5

Subtract $5$ from $- 9$ .

$- 2 x^{2} + 16 x - 14 = 0$

Step 5.6

Factor the left side of the equation.

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Step 5.6.1

Factor $- 2$ out of $- 2 x^{2} + 16 x - 14$ .

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Step 5.6.1.1

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} + 16 x - 14 = 0$

Step 5.6.1.2

Factor $- 2$ out of $16 x$ .

$- 2 x^{2} - 2 (- 8 x) - 14 = 0$

Step 5.6.1.3

Factor $- 2$ out of $- 14$ .

$- 2 x^{2} - 2 (- 8 x) - 2 \cdot 7 = 0$

Step 5.6.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 8 x)$ .

$- 2 (x^{2} - 8 x) - 2 \cdot 7 = 0$

Step 5.6.1.5

Factor $- 2$ out of $- 2 (x^{2} - 8 x) - 2 (7)$ .

$- 2 (x^{2} - 8 x + 7) = 0$

Step 5.6.2

Factor.

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Step 5.6.2.1

Factor $x^{2} - 8 x + 7$ using the AC method.

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Step 5.6.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $- 8$ .

$- 7, - 1$

Step 5.6.2.1.2

Write the factored form using these integers.

$- 2 ((x - 7) (x - 1)) = 0$

Step 5.6.2.2

Remove unnecessary parentheses.

$- 2 (x - 7) (x - 1) = 0$

Step 5.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 7 = 0$

$x - 1 = 0$

Step 5.8

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 5.8.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 5.8.2

Add $7$ to both sides of the equation.

$x = 7$

Step 5.9

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 5.9.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 5.9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.10

The final solution is all the values that make $- 2 (x - 7) (x - 1) = 0$ true.

$x = 7, 1$

Step 6

List all of the solutions.

$x = 2, 1, 7, 1$

Step 7

Use each root to create test intervals.

$x < 1$

$1 < x < 2$

$2 < x < 7$

$x > 7$

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 $x < 1$ to see if it makes the inequality true.

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Step 8.1.1

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 8.1.2

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

$| (- 2 \cdot 0 + 9) ((0) - 1) | < 5 | (0) - 1 |$

Step 8.1.3

The left side $9$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $1 < x < 2$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $1 < x < 2$ and see if this value makes the original inequality true.

$x = 1.5$

Step 8.2.2

Replace $x$ with $1.5$ in the original inequality.

$| (- 2 \cdot 1.5 + 9) ((1.5) - 1) | < 5 | (1.5) - 1 |$

Step 8.2.3

The left side $3$ is not less than the right side $2.5$ , which means that the given statement is false.

False

Step 8.3

Test a value on the interval $2 < x < 7$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $2 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Step 8.3.2

Replace $x$ with $4$ in the original inequality.

$| (- 2 \cdot 4 + 9) ((4) - 1) | < 5 | (4) - 1 |$

Step 8.3.3

The left side $3$ is less than the right side $15$ , which means that the given statement is always true.

True

Step 8.4

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 8.4.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 8.4.2

Replace $x$ with $10$ in the original inequality.

$| (- 2 \cdot 10 + 9) ((10) - 1) | < 5 | (10) - 1 |$

Step 8.4.3

The left side $99$ is not less than the right side $45$ , which means that the given statement is false.

False

Step 8.5

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

$x < 1$ False

$1 < x < 2$ False

$2 < x < 7$ True

$x > 7$ False

$x < 1$ False

$1 < x < 2$ False

$2 < x < 7$ True

$x > 7$ False

Step 9

The solution consists of all of the true intervals.

$2 < x < 7$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 7$

Interval Notation:

$(2, 7)$

Step 11