Algebra Examples

$0 < x + 1 \leq 3 (x - 1)$

Step 1

Solve $0 < x + 1$ for $x$ .

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Step 1.1

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

$x + 1 > 0$

Step 1.2

Subtract $1$ from both sides of the inequality.

$x > - 1$

Step 2

Solve $x + 1 \leq 3 (x - 1)$ for $x$ .

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Step 2.1

Simplify $3 (x - 1)$ .

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Step 2.1.1

Rewrite.

$x + 1 \leq 0 + 0 + 3 (x - 1)$

Step 2.1.2

Simplify by adding zeros.

$x + 1 \leq 3 (x - 1)$

Step 2.1.3

Apply the distributive property.

$x + 1 \leq 3 x + 3 \cdot - 1$

Step 2.1.4

Multiply $3$ by $- 1$ .

$x + 1 \leq 3 x - 3$

Step 2.2

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

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Step 2.2.1

Subtract $3 x$ from both sides of the inequality.

$x + 1 - 3 x \leq - 3$

Step 2.2.2

Subtract $3 x$ from $x$ .

$- 2 x + 1 \leq - 3$

Step 2.3

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

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Step 2.3.1

Subtract $1$ from both sides of the inequality.

$- 2 x \leq - 3 - 1$

Step 2.3.2

Subtract $1$ from $- 3$ .

$- 2 x \leq - 4$

Step 2.4

Divide each term in $- 2 x \leq - 4$ by $- 2$ and simplify.

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Step 2.4.1

Divide each term in $- 2 x \leq - 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{- 4}{- 2}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{- 4}{- 2}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 4}{- 2}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 4$ by $- 2$ .

$x \geq 2$

Step 3

The result can be shown in multiple forms.

Inequality Form:

$x \geq 2$

Interval Notation:

$[2, \infty)$

Step 4