Algebra Examples

$6 (x^{2} + 1) - (2 x - 4) (3 x + 2) < 3 (5 x + 21)$

Step 1

Simplify $6 (x^{2} + 1) - (2 x - 4) (3 x + 2)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$6 x^{2} + 6 \cdot 1 - (2 x - 4) (3 x + 2) < 3 (5 x + 21)$

Step 1.1.2

Multiply $6$ by $1$ .

$6 x^{2} + 6 - (2 x - 4) (3 x + 2) < 3 (5 x + 21)$

Step 1.1.3

Apply the distributive property.

$6 x^{2} + 6 + (- (2 x) - - 4) (3 x + 2) < 3 (5 x + 21)$

Step 1.1.4

Multiply $2$ by $- 1$ .

$6 x^{2} + 6 + (- 2 x - - 4) (3 x + 2) < 3 (5 x + 21)$

Step 1.1.5

Multiply $- 1$ by $- 4$ .

$6 x^{2} + 6 + (- 2 x + 4) (3 x + 2) < 3 (5 x + 21)$

Step 1.1.6

Expand $(- 2 x + 4) (3 x + 2)$ using the FOIL Method.

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Step 1.1.6.1

Apply the distributive property.

$6 x^{2} + 6 - 2 x (3 x + 2) + 4 (3 x + 2) < 3 (5 x + 21)$

Step 1.1.6.2

Apply the distributive property.

$6 x^{2} + 6 - 2 x (3 x) - 2 x \cdot 2 + 4 (3 x + 2) < 3 (5 x + 21)$

Step 1.1.6.3

Apply the distributive property.

$6 x^{2} + 6 - 2 x (3 x) - 2 x \cdot 2 + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7

Simplify and combine like terms.

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Step 1.1.7.1

Simplify each term.

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Step 1.1.7.1.1

Rewrite using the commutative property of multiplication.

$6 x^{2} + 6 - 2 \cdot 3 x \cdot x - 2 x \cdot 2 + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.2

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

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Step 1.1.7.1.2.1

Move $x$ .

$6 x^{2} + 6 - 2 \cdot 3 (x \cdot x) - 2 x \cdot 2 + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.2.2

Multiply $x$ by $x$ .

$6 x^{2} + 6 - 2 \cdot 3 x^{2} - 2 x \cdot 2 + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.3

Multiply $- 2$ by $3$ .

$6 x^{2} + 6 - 6 x^{2} - 2 x \cdot 2 + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.4

Multiply $2$ by $- 2$ .

$6 x^{2} + 6 - 6 x^{2} - 4 x + 4 (3 x) + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.5

Multiply $3$ by $4$ .

$6 x^{2} + 6 - 6 x^{2} - 4 x + 12 x + 4 \cdot 2 < 3 (5 x + 21)$

Step 1.1.7.1.6

Multiply $4$ by $2$ .

$6 x^{2} + 6 - 6 x^{2} - 4 x + 12 x + 8 < 3 (5 x + 21)$

Step 1.1.7.2

Add $- 4 x$ and $12 x$ .

$6 x^{2} + 6 - 6 x^{2} + 8 x + 8 < 3 (5 x + 21)$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Combine the opposite terms in $6 x^{2} + 6 - 6 x^{2} + 8 x + 8$ .

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Step 1.2.1.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$0 + 6 + 8 x + 8 < 3 (5 x + 21)$

Step 1.2.1.2

Add $0$ and $6$ .

$6 + 8 x + 8 < 3 (5 x + 21)$

Step 1.2.2

Add $6$ and $8$ .

$8 x + 14 < 3 (5 x + 21)$

Step 2

Simplify $3 (5 x + 21)$ .

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Step 2.1

Apply the distributive property.

$8 x + 14 < 3 (5 x) + 3 \cdot 21$

Step 2.2

Multiply.

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Step 2.2.1

Multiply $5$ by $3$ .

$8 x + 14 < 15 x + 3 \cdot 21$

Step 2.2.2

Multiply $3$ by $21$ .

$8 x + 14 < 15 x + 63$

Step 3

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

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Step 3.1

Subtract $15 x$ from both sides of the inequality.

$8 x + 14 - 15 x < 63$

Step 3.2

Subtract $15 x$ from $8 x$ .

$- 7 x + 14 < 63$

Step 4

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.1

Subtract $14$ from both sides of the inequality.

$- 7 x < 63 - 14$

Step 4.2

Subtract $14$ from $63$ .

$- 7 x < 49$

Step 5

Divide each term in $- 7 x < 49$ by $- 7$ and simplify.

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Step 5.1

Divide each term in $- 7 x < 49$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} > \frac{49}{- 7}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 7$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} > \frac{49}{- 7}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x > \frac{49}{- 7}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $49$ by $- 7$ .

$x > - 7$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > - 7$

Interval Notation:

$(- 7, \infty)$

Step 7