Solve the Inequality for x (x-4)(3x+1)<(2x-6)(x-2)+4

$(x - 4) (3 x + 1) < (2 x - 6) (x - 2) + 4$

Step 1

Simplify $(x - 4) (3 x + 1)$ .

Rewrite.

$0 + 0 + (x - 4) (3 x + 1) < (2 x - 6) (x - 2) + 4$

Simplify by adding zeros.

$(x - 4) (3 x + 1) < (2 x - 6) (x - 2) + 4$

Expand $(x - 4) (3 x + 1)$ using the FOIL Method.

Apply the distributive property.

$x (3 x + 1) - 4 (3 x + 1) < (2 x - 6) (x - 2) + 4$

Apply the distributive property.

$x (3 x) + x \cdot 1 - 4 (3 x + 1) < (2 x - 6) (x - 2) + 4$

Apply the distributive property.

$x (3 x) + x \cdot 1 - 4 (3 x) - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 1 - 4 (3 x) - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

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

Move $x$ .

$3 (x \cdot x) + x \cdot 1 - 4 (3 x) - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

Multiply $x$ by $x$ .

$3 x^{2} + x \cdot 1 - 4 (3 x) - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

Multiply $x$ by $1$ .

$3 x^{2} + x - 4 (3 x) - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

Multiply $3$ by $- 4$ .

$3 x^{2} + x - 12 x - 4 \cdot 1 < (2 x - 6) (x - 2) + 4$

Multiply $- 4$ by $1$ .

$3 x^{2} + x - 12 x - 4 < (2 x - 6) (x - 2) + 4$

Subtract $12 x$ from $x$ .

$3 x^{2} - 11 x - 4 < (2 x - 6) (x - 2) + 4$

Step 2

Simplify $(2 x - 6) (x - 2) + 4$ .

Simplify each term.

Expand $(2 x - 6) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$3 x^{2} - 11 x - 4 < 2 x (x - 2) - 6 (x - 2) + 4$

Apply the distributive property.

$3 x^{2} - 11 x - 4 < 2 x \cdot x + 2 x \cdot - 2 - 6 (x - 2) + 4$

Apply the distributive property.

$3 x^{2} - 11 x - 4 < 2 x \cdot x + 2 x \cdot - 2 - 6 x - 6 \cdot - 2 + 4$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$3 x^{2} - 11 x - 4 < 2 (x \cdot x) + 2 x \cdot - 2 - 6 x - 6 \cdot - 2 + 4$

Multiply $x$ by $x$ .

$3 x^{2} - 11 x - 4 < 2 x^{2} + 2 x \cdot - 2 - 6 x - 6 \cdot - 2 + 4$

Multiply $- 2$ by $2$ .

$3 x^{2} - 11 x - 4 < 2 x^{2} - 4 x - 6 x - 6 \cdot - 2 + 4$

Multiply $- 6$ by $- 2$ .

$3 x^{2} - 11 x - 4 < 2 x^{2} - 4 x - 6 x + 12 + 4$

Subtract $6 x$ from $- 4 x$ .

$3 x^{2} - 11 x - 4 < 2 x^{2} - 10 x + 12 + 4$

Add $12$ and $4$ .

$3 x^{2} - 11 x - 4 < 2 x^{2} - 10 x + 16$

Step 3

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

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

$3 x^{2} - 11 x - 4 - 2 x^{2} < - 10 x + 16$

Add $10 x$ to both sides of the inequality.

$3 x^{2} - 11 x - 4 - 2 x^{2} + 10 x < 16$

Subtract $2 x^{2}$ from $3 x^{2}$ .

$x^{2} - 11 x - 4 + 10 x < 16$

Add $- 11 x$ and $10 x$ .

$x^{2} - x - 4 < 16$

Step 4

Convert the inequality to an equation.

$x^{2} - x - 4 = 16$

Step 5

Subtract $16$ from both sides of the equation.

$x^{2} - x - 4 - 16 = 0$

Step 6

Subtract $16$ from $- 4$ .

$x^{2} - x - 20 = 0$

Step 7

Factor $x^{2} - x - 20$ using the AC method.

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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(x - 5) (x + 4) = 0$

Step 8

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

$x - 5 = 0$

$x + 4 = 0$

Step 9

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 10

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11

The final solution is all the values that make $(x - 5) (x + 4) = 0$ true.

$x = 5, - 4$

Step 12

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 5$

$x > 5$

Step 13

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

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

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

$((- 6) - 4) (3 (- 6) + 1) < (2 (- 6) - 6) ((- 6) - 2) + 4$

The left side $170$ is not less than the right side $148$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$((0) - 4) (3 (0) + 1) < (2 (0) - 6) ((0) - 2) + 4$

The left side $- 4$ is less than the right side $16$ , which means that the given statement is always true.

True

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$((8) - 4) (3 (8) + 1) < (2 (8) - 6) ((8) - 2) + 4$

The left side $100$ is not less than the right side $64$ , which means that the given statement is false.

False

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

$x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

$x < - 4$ False

$- 4 < x < 5$ True

$x > 5$ False

Step 14

The solution consists of all of the true intervals.

$- 4 < x < 5$

Step 15

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 5$

Interval Notation:

$(- 4, 5)$

Step 16

Solve the Inequality for x (x-4)(3x+1)<(2x-6)(x-2)+4

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