Precalculus Examples

$4 x (x - 2) < (2 x - 1) (x - 3)$

Step 1

Simplify $4 x (x - 2)$ .

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Step 1.1

Rewrite.

$0 + 0 + 4 x (x - 2) < (2 x - 1) (x - 3)$

Step 1.2

Simplify by adding zeros.

$4 x (x - 2) < (2 x - 1) (x - 3)$

Step 1.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 2 < (2 x - 1) (x - 3)$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 2 < (2 x - 1) (x - 3)$

Step 1.4.2

Multiply $x$ by $x$ .

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

Step 1.5

Multiply $- 2$ by $4$ .

$4 x^{2} - 8 x < (2 x - 1) (x - 3)$

Step 2

Simplify $(2 x - 1) (x - 3)$ .

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Step 2.1

Expand $(2 x - 1) (x - 3)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$4 x^{2} - 8 x < 2 x (x - 3) - 1 (x - 3)$

Step 2.1.2

Apply the distributive property.

$4 x^{2} - 8 x < 2 x \cdot x + 2 x \cdot - 3 - 1 (x - 3)$

Step 2.1.3

Apply the distributive property.

$4 x^{2} - 8 x < 2 x \cdot x + 2 x \cdot - 3 - 1 x - 1 \cdot - 3$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Move $x$ .

$4 x^{2} - 8 x < 2 (x \cdot x) + 2 x \cdot - 3 - 1 x - 1 \cdot - 3$

Step 2.2.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} - 8 x < 2 x^{2} + 2 x \cdot - 3 - 1 x - 1 \cdot - 3$

Step 2.2.1.2

Multiply $- 3$ by $2$ .

$4 x^{2} - 8 x < 2 x^{2} - 6 x - 1 x - 1 \cdot - 3$

Step 2.2.1.3

Rewrite $- 1 x$ as $- x$ .

$4 x^{2} - 8 x < 2 x^{2} - 6 x - x - 1 \cdot - 3$

Step 2.2.1.4

Multiply $- 1$ by $- 3$ .

$4 x^{2} - 8 x < 2 x^{2} - 6 x - x + 3$

Step 2.2.2

Subtract $x$ from $- 6 x$ .

$4 x^{2} - 8 x < 2 x^{2} - 7 x + 3$

Step 3

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

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Step 3.1

Subtract $2 x^{2}$ from both sides of the inequality.

$4 x^{2} - 8 x - 2 x^{2} < - 7 x + 3$

Step 3.2

Add $7 x$ to both sides of the inequality.

$4 x^{2} - 8 x - 2 x^{2} + 7 x < 3$

Step 3.3

Subtract $2 x^{2}$ from $4 x^{2}$ .

$2 x^{2} - 8 x + 7 x < 3$

Step 3.4

Add $- 8 x$ and $7 x$ .

$2 x^{2} - x < 3$

Step 4

Convert the inequality to an equation.

$2 x^{2} - x = 3$

Step 5

Subtract $3$ from both sides of the equation.

$2 x^{2} - x - 3 = 0$

Step 6

Factor by grouping.

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Step 6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 6.1.1

Factor $- 1$ out of $- x$ .

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

Step 6.1.2

Rewrite $- 1$ as $2$ plus $- 3$

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

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

Step 6.2.2

Factor out the greatest common factor (GCF) from each group.

$2 x (x + 1) - 3 (x + 1) = 0$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$(x + 1) (2 x - 3) = 0$

Step 7

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

$x + 1 = 0$

$2 x - 3 = 0$

Step 8

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

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Step 8.1

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

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 9.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 9.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 9.2.2

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

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Step 9.2.2.1

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

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

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 10

The final solution is all the values that make $(x + 1) (2 x - 3) = 0$ true.

$x = - 1, \frac{3}{2}$

Step 11

Use each root to create test intervals.

$x < - 1$

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

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

Step 12

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 12.1

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

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Step 12.1.1

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

$x = - 4$

Step 12.1.2

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

$4 (- 4) ((- 4) - 2) < (2 (- 4) - 1) ((- 4) - 3)$

Step 12.1.3

The left side $96$ is not less than the right side $63$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- 1 < x < \frac{3}{2}$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- 1 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 12.2.2

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

$4 (0) ((0) - 2) < (2 (0) - 1) ((0) - 3)$

Step 12.2.3

The left side $0$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $x > \frac{3}{2}$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $x > \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 4$

Step 12.3.2

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

$4 (4) ((4) - 2) < (2 (4) - 1) ((4) - 3)$

Step 12.3.3

The left side $32$ is not less than the right side $7$ , which means that the given statement is false.

False

Step 12.4

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

$x < - 1$ False

$- 1 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

$x < - 1$ False

$- 1 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

Step 13

The solution consists of all of the true intervals.

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

Step 14

Convert the inequality to interval notation.

$(- 1, \frac{3}{2})$

Step 15