Trigonometry Examples

$4 x (x - 8) < 2 (2 x - 1) (x - 9)$

Step 1

Simplify $4 x (x - 8)$ .

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Step 1.1

Rewrite.

$0 + 0 + 4 x (x - 8) < 2 (2 x - 1) (x - 9)$

Step 1.2

Simplify by adding zeros.

$4 x (x - 8) < 2 (2 x - 1) (x - 9)$

Step 1.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 8 < 2 (2 x - 1) (x - 9)$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 8 < 2 (2 x - 1) (x - 9)$

Step 1.4.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot - 8 < 2 (2 x - 1) (x - 9)$

Step 1.5

Multiply $- 8$ by $4$ .

$4 x^{2} - 32 x < 2 (2 x - 1) (x - 9)$

Step 2

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

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$4 x^{2} - 32 x < (2 (2 x) + 2 \cdot - 1) (x - 9)$

Step 2.1.2

Multiply.

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Step 2.1.2.1

Multiply $2$ by $2$ .

$4 x^{2} - 32 x < (4 x + 2 \cdot - 1) (x - 9)$

Step 2.1.2.2

Multiply $2$ by $- 1$ .

$4 x^{2} - 32 x < (4 x - 2) (x - 9)$

Step 2.2

Expand $(4 x - 2) (x - 9)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$4 x^{2} - 32 x < 4 x (x - 9) - 2 (x - 9)$

Step 2.2.2

Apply the distributive property.

$4 x^{2} - 32 x < 4 x \cdot x + 4 x \cdot - 9 - 2 (x - 9)$

Step 2.2.3

Apply the distributive property.

$4 x^{2} - 32 x < 4 x \cdot x + 4 x \cdot - 9 - 2 x - 2 \cdot - 9$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Move $x$ .

$4 x^{2} - 32 x < 4 (x \cdot x) + 4 x \cdot - 9 - 2 x - 2 \cdot - 9$

Step 2.3.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} - 32 x < 4 x^{2} + 4 x \cdot - 9 - 2 x - 2 \cdot - 9$

Step 2.3.1.2

Multiply $- 9$ by $4$ .

$4 x^{2} - 32 x < 4 x^{2} - 36 x - 2 x - 2 \cdot - 9$

Step 2.3.1.3

Multiply $- 2$ by $- 9$ .

$4 x^{2} - 32 x < 4 x^{2} - 36 x - 2 x + 18$

Step 2.3.2

Subtract $2 x$ from $- 36 x$ .

$4 x^{2} - 32 x < 4 x^{2} - 38 x + 18$

Step 3

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

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Step 3.1

Subtract $4 x^{2}$ from both sides of the inequality.

$4 x^{2} - 32 x - 4 x^{2} < - 38 x + 18$

Step 3.2

Add $38 x$ to both sides of the inequality.

$4 x^{2} - 32 x - 4 x^{2} + 38 x < 18$

Step 3.3

Combine the opposite terms in $4 x^{2} - 32 x - 4 x^{2} + 38 x$ .

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Step 3.3.1

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

$- 32 x + 0 + 38 x < 18$

Step 3.3.2

Add $- 32 x$ and $0$ .

$- 32 x + 38 x < 18$

Step 3.4

Add $- 32 x$ and $38 x$ .

$6 x < 18$

Step 4

Divide each term in $6 x < 18$ by $6$ and simplify.

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Step 4.1

Divide each term in $6 x < 18$ by $6$ .

$\frac{6 x}{6} < \frac{18}{6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} < \frac{18}{6}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x < \frac{18}{6}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $18$ by $6$ .

$x < 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 3$

Interval Notation:

$(- \infty, 3)$

Step 6