Algebra Examples

$- 2 (x - 1) > \frac{y}{2} + 4$ $\frac{x - 1}{3} \leq \frac{x - 4}{6}$

Step 1

Graph $- 2 (x - 1) > \frac{y}{2} + 4$ .

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Step 1.1

Solve for $y$ .

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Step 1.1.1

Rewrite so $y$ is on the left side of the inequality.

$\frac{y}{2} + 4 < - 2 (x - 1)$

Step 1.1.2

Simplify $- 2 (x - 1)$ .

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Step 1.1.2.1

Rewrite.

$\frac{y}{2} + 4 < 0 + 0 - 2 (x - 1)$

Step 1.1.2.2

Simplify by adding zeros.

$\frac{y}{2} + 4 < - 2 (x - 1)$

Step 1.1.2.3

Apply the distributive property.

$\frac{y}{2} + 4 < - 2 x - 2 \cdot - 1$

Step 1.1.2.4

Multiply $- 2$ by $- 1$ .

$\frac{y}{2} + 4 < - 2 x + 2$

Step 1.1.3

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

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Step 1.1.3.1

Subtract $4$ from both sides of the inequality.

$\frac{y}{2} < - 2 x + 2 - 4$

Step 1.1.3.2

Subtract $4$ from $2$ .

$\frac{y}{2} < - 2 x - 2$

Step 1.1.4

Multiply both sides by $2$ .

$\frac{y}{2} \cdot 2 < (- 2 x - 2) \cdot 2$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Simplify the left side.

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Step 1.1.5.1.1

Cancel the common factor of $2$ .

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Step 1.1.5.1.1.1

Cancel the common factor.

$\frac{y}{2} \cdot 2 < (- 2 x - 2) \cdot 2$

Step 1.1.5.1.1.2

Rewrite the expression.

$y < (- 2 x - 2) \cdot 2$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify $(- 2 x - 2) \cdot 2$ .

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Step 1.1.5.2.1.1

Apply the distributive property.

$y < - 2 x \cdot 2 - 2 \cdot 2$

Step 1.1.5.2.1.2

Multiply.

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Step 1.1.5.2.1.2.1

Multiply $2$ by $- 2$ .

$y < - 4 x - 2 \cdot 2$

Step 1.1.5.2.1.2.2

Multiply $- 2$ by $2$ .

$y < - 4 x - 4$

Step 1.2

Use the slope-intercept form to find the slope and y-intercept.

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Step 1.2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = - 4$

$b = - 4$

Step 1.2.3

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- 4$

y-intercept: $(0, - 4)$

Slope: $- 4$

y-intercept: $(0, - 4)$

Step 1.3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $- 4 x - 4$ .

$y < - 4 x - 4$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides by $3$ .

$\frac{x - 1}{3} \cdot 3 \leq \frac{x - 4}{6} \cdot 3$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x - 1}{3} \cdot 3 \leq \frac{x - 4}{6} \cdot 3$

Step 2.2.1.1.2

Rewrite the expression.

$x - 1 \leq \frac{x - 4}{6} \cdot 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Factor $3$ out of $6$ .

$x - 1 \leq \frac{x - 4}{3 (2)} \cdot 3$

Step 2.2.2.1.2

Cancel the common factor.

$x - 1 \leq \frac{x - 4}{3 \cdot 2} \cdot 3$

Step 2.2.2.1.3

Rewrite the expression.

$x - 1 \leq \frac{x - 4}{2}$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Multiply both sides by $2$ .

$(x - 1) \cdot 2 \leq \frac{x - 4}{2} \cdot 2$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Simplify $(x - 1) \cdot 2$ .

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Step 2.3.2.1.1.1

Apply the distributive property.

$x \cdot 2 - 1 \cdot 2 \leq \frac{x - 4}{2} \cdot 2$

Step 2.3.2.1.1.2

Simplify the expression.

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Step 2.3.2.1.1.2.1

Move $2$ to the left of $x$ .

$2 \cdot x - 1 \cdot 2 \leq \frac{x - 4}{2} \cdot 2$

Step 2.3.2.1.1.2.2

Multiply $- 1$ by $2$ .

$2 x - 2 \leq \frac{x - 4}{2} \cdot 2$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$2 x - 2 \leq \frac{x - 4}{2} \cdot 2$

Step 2.3.2.2.1.2

Rewrite the expression.

$2 x - 2 \leq x - 4$

Step 2.3.3

Solve for $x$ .

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Step 2.3.3.1

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

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Step 2.3.3.1.1

Subtract $x$ from both sides of the inequality.

$2 x - 2 - x \leq - 4$

Step 2.3.3.1.2

Subtract $x$ from $2 x$ .

$x - 2 \leq - 4$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Add $2$ to both sides of the inequality.

$x \leq - 4 + 2$

Step 2.3.3.2.2

Add $- 4$ and $2$ .

$x \leq - 2$

Step 3

Plot each graph on the same coordinate system.

$- 2 (x - 1) > \frac{y}{2} + 4$

$\frac{x - 1}{3} \leq \frac{x - 4}{6}$

Step 4