Algebra Examples

${(x - 3)}^{2} + 5 x \geq {(1 - x)}^{2} + 3 x$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(1 - x)}^{2}$ as $(1 - x) (1 - x)$ .

${(x - 3)}^{2} + 5 x \geq (1 - x) (1 - x) + 3 x$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

${(x - 3)}^{2} + 5 x \geq 1 (1 - x) - x (1 - x) + 3 x$

Step 1.1.2.2

Apply the distributive property.

${(x - 3)}^{2} + 5 x \geq 1 \cdot 1 + 1 (- x) - x (1 - x) + 3 x$

Step 1.1.2.3

Apply the distributive property.

${(x - 3)}^{2} + 5 x \geq 1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x) + 3 x$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $1$ by $1$ .

${(x - 3)}^{2} + 5 x \geq 1 + 1 (- x) - x \cdot 1 - x (- x) + 3 x$

Step 1.1.3.1.2

Multiply $- x$ by $1$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x \cdot 1 - x (- x) + 3 x$

Step 1.1.3.1.3

Multiply $- 1$ by $1$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x - x (- x) + 3 x$

Step 1.1.3.1.4

Rewrite using the commutative property of multiplication.

${(x - 3)}^{2} + 5 x \geq 1 - x - x - 1 \cdot - 1 x \cdot x + 3 x$

Step 1.1.3.1.5

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

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Step 1.1.3.1.5.1

Move $x$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x - 1 \cdot - 1 (x \cdot x) + 3 x$

Step 1.1.3.1.5.2

Multiply $x$ by $x$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x - 1 \cdot - 1 x^{2} + 3 x$

Step 1.1.3.1.6

Multiply $- 1$ by $- 1$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x + 1 x^{2} + 3 x$

Step 1.1.3.1.7

Multiply $x^{2}$ by $1$ .

${(x - 3)}^{2} + 5 x \geq 1 - x - x + x^{2} + 3 x$

Step 1.1.3.2

Subtract $x$ from $- x$ .

${(x - 3)}^{2} + 5 x \geq 1 - 2 x + x^{2} + 3 x$

Step 1.2

Add $- 2 x$ and $3 x$ .

${(x - 3)}^{2} + 5 x \geq 1 + x^{2} + x$

Step 2

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

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Step 2.1

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

${(x - 3)}^{2} + 5 x - x^{2} \geq 1 + x$

Step 2.2

Subtract $x$ from both sides of the inequality.

${(x - 3)}^{2} + 5 x - x^{2} - x \geq 1$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$(x - 3) (x - 3) + 5 x - x^{2} - x \geq 1$

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

$x (x - 3) - 3 (x - 3) + 5 x - x^{2} - x \geq 1$

Step 2.3.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) + 5 x - x^{2} - x \geq 1$

Step 2.3.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + 5 x - x^{2} - x \geq 1$

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + 5 x - x^{2} - x \geq 1$

Step 2.3.3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + 5 x - x^{2} - x \geq 1$

Step 2.3.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 + 5 x - x^{2} - x \geq 1$

Step 2.3.3.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 + 5 x - x^{2} - x \geq 1$

Step 2.4

Combine the opposite terms in $x^{2} - 6 x + 9 + 5 x - x^{2} - x$ .

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Step 2.4.1

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

$- 6 x + 9 + 5 x + 0 - x \geq 1$

Step 2.4.2

Add $- 6 x + 9 + 5 x$ and $0$ .

$- 6 x + 9 + 5 x - x \geq 1$

Step 2.5

Add $- 6 x$ and $5 x$ .

$- x + 9 - x \geq 1$

Step 2.6

Subtract $x$ from $- x$ .

$- 2 x + 9 \geq 1$

Step 3

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

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Step 3.1

Subtract $9$ from both sides of the inequality.

$- 2 x \geq 1 - 9$

Step 3.2

Subtract $9$ from $1$ .

$- 2 x \geq - 8$

Step 4

Divide each term in $- 2 x \geq - 8$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 x \geq - 8$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 8}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 8}{- 2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 8}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 8$ by $- 2$ .

$x \leq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq 4$

Interval Notation:

$(- \infty, 4]$

Step 6