Algebra Examples

$10 x (x - 5) < 2 (5 x - 1) (x - 6)$

Step 1

Solve $x < 1$ .

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Step 1.1

Simplify $10 x (x - 5)$ .

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Step 1.1.1

Rewrite.

$0 + 0 + 10 x (x - 5) < 2 (5 x - 1) (x - 6)$

Step 1.1.2

Simplify by adding zeros.

$10 x (x - 5) < 2 (5 x - 1) (x - 6)$

Step 1.1.3

Apply the distributive property.

$10 x \cdot x + 10 x \cdot - 5 < 2 (5 x - 1) (x - 6)$

Step 1.1.4

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

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Step 1.1.4.1

Move $x$ .

$10 (x \cdot x) + 10 x \cdot - 5 < 2 (5 x - 1) (x - 6)$

Step 1.1.4.2

Multiply $x$ by $x$ .

$10 x^{2} + 10 x \cdot - 5 < 2 (5 x - 1) (x - 6)$

Step 1.1.5

Multiply $- 5$ by $10$ .

$10 x^{2} - 50 x < 2 (5 x - 1) (x - 6)$

Step 1.2

Simplify $2 (5 x - 1) (x - 6)$ .

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Step 1.2.1

Simplify by multiplying through.

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Step 1.2.1.1

Apply the distributive property.

$10 x^{2} - 50 x < (2 (5 x) + 2 \cdot - 1) (x - 6)$

Step 1.2.1.2

Multiply.

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Step 1.2.1.2.1

Multiply $5$ by $2$ .

$10 x^{2} - 50 x < (10 x + 2 \cdot - 1) (x - 6)$

Step 1.2.1.2.2

Multiply $2$ by $- 1$ .

$10 x^{2} - 50 x < (10 x - 2) (x - 6)$

Step 1.2.2

Expand $(10 x - 2) (x - 6)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$10 x^{2} - 50 x < 10 x (x - 6) - 2 (x - 6)$

Step 1.2.2.2

Apply the distributive property.

$10 x^{2} - 50 x < 10 x \cdot x + 10 x \cdot - 6 - 2 (x - 6)$

Step 1.2.2.3

Apply the distributive property.

$10 x^{2} - 50 x < 10 x \cdot x + 10 x \cdot - 6 - 2 x - 2 \cdot - 6$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

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Step 1.2.3.1.1.1

Move $x$ .

$10 x^{2} - 50 x < 10 (x \cdot x) + 10 x \cdot - 6 - 2 x - 2 \cdot - 6$

Step 1.2.3.1.1.2

Multiply $x$ by $x$ .

$10 x^{2} - 50 x < 10 x^{2} + 10 x \cdot - 6 - 2 x - 2 \cdot - 6$

Step 1.2.3.1.2

Multiply $- 6$ by $10$ .

$10 x^{2} - 50 x < 10 x^{2} - 60 x - 2 x - 2 \cdot - 6$

Step 1.2.3.1.3

Multiply $- 2$ by $- 6$ .

$10 x^{2} - 50 x < 10 x^{2} - 60 x - 2 x + 12$

Step 1.2.3.2

Subtract $2 x$ from $- 60 x$ .

$10 x^{2} - 50 x < 10 x^{2} - 62 x + 12$

Step 1.3

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

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Step 1.3.1

Subtract $10 x^{2}$ from both sides of the inequality.

$10 x^{2} - 50 x - 10 x^{2} < - 62 x + 12$

Step 1.3.2

Add $62 x$ to both sides of the inequality.

$10 x^{2} - 50 x - 10 x^{2} + 62 x < 12$

Step 1.3.3

Combine the opposite terms in $10 x^{2} - 50 x - 10 x^{2} + 62 x$ .

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Step 1.3.3.1

Subtract $10 x^{2}$ from $10 x^{2}$ .

$- 50 x + 0 + 62 x < 12$

Step 1.3.3.2

Add $- 50 x$ and $0$ .

$- 50 x + 62 x < 12$

Step 1.3.4

Add $- 50 x$ and $62 x$ .

$12 x < 12$

Step 1.4

Divide each term in $12 x < 12$ by $12$ and simplify.

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Step 1.4.1

Divide each term in $12 x < 12$ by $12$ .

$\frac{12 x}{12} < \frac{12}{12}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Cancel the common factor of $12$ .

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Step 1.4.2.1.1

Cancel the common factor.

$\frac{12 x}{12} < \frac{12}{12}$

Step 1.4.2.1.2

Divide $x$ by $1$ .

$x < \frac{12}{12}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $12$ by $12$ .

$x < 1$

Step 2

Use the inequality $x < 1$ to build the set notation.

${x | x < 1}$

Step 3