Algebra Examples

$y < {(x - 1)}^{2} - 3$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$y < (x - 1) (x - 1) - 3$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$y < x (x - 1) - 1 (x - 1) - 3$

Step 1.1.2.2

Apply the distributive property.

$y < x \cdot x + x \cdot - 1 - 1 (x - 1) - 3$

Step 1.1.2.3

Apply the distributive property.

$y < x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 3$

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 $x$ by $x$ .

$y < x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 3$

Step 1.1.3.1.2

Move $- 1$ to the left of $x$ .

$y < x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 3$

Step 1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$y < x^{2} - x - 1 x - 1 \cdot - 1 - 3$

Step 1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$y < x^{2} - x - x - 1 \cdot - 1 - 3$

Step 1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$y < x^{2} - x - x + 1 - 3$

Step 1.1.3.2

Subtract $x$ from $- x$ .

$y < x^{2} - 2 x + 1 - 3$

Step 1.2

Subtract $3$ from $1$ .

$y < x^{2} - 2 x - 2$

Step 2

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $x^{2} - 2 x - 2$ .

$y < x^{2} - 2 x - 2$

Step 4