Algebra Examples

$- 2.5 \leq \frac{1 - 3 y}{2} \leq 1.5$

Step 1

Solve $- 2.5 \leq \frac{1 - 3 y}{2}$ for $y$ .

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Step 1.1

Multiply both sides by $2$ .

$- 2.5 \cdot 2 \leq \frac{1 - 3 y}{2} \cdot 2$

Step 1.2

Simplify.

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Multiply $- 2.5$ by $2$ .

$- 5 \leq \frac{1 - 3 y}{2} \cdot 2$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Simplify $\frac{1 - 3 y}{2} \cdot 2$ .

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Step 1.2.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$- 5 \leq \frac{1 - 3 y}{2} \cdot 2$

Step 1.2.2.1.1.2

Rewrite the expression.

$- 5 \leq 1 - 3 y$

Step 1.2.2.1.2

Reorder $1$ and $- 3 y$ .

$- 5 \leq - 3 y + 1$

Step 1.3

Solve for $y$ .

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Step 1.3.1

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

$- 3 y + 1 \geq - 5$

Step 1.3.2

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

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Step 1.3.2.1

Subtract $1$ from both sides of the inequality.

$- 3 y \geq - 5 - 1$

Step 1.3.2.2

Subtract $1$ from $- 5$ .

$- 3 y \geq - 6$

Step 1.3.3

Divide each term in $- 3 y \geq - 6$ by $- 3$ and simplify.

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Step 1.3.3.1

Divide each term in $- 3 y \geq - 6$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 y}{- 3} \leq \frac{- 6}{- 3}$

Step 1.3.3.2

Simplify the left side.

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Step 1.3.3.2.1

Cancel the common factor of $- 3$ .

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Step 1.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} \leq \frac{- 6}{- 3}$

Step 1.3.3.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- 6}{- 3}$

Step 1.3.3.3

Simplify the right side.

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Step 1.3.3.3.1

Divide $- 6$ by $- 3$ .

$y \leq 2$

Step 2

Solve $\frac{1 - 3 y}{2} \leq 1.5$ for $y$ .

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Step 2.1

Multiply both sides by $2$ .

$\frac{1 - 3 y}{2} \cdot 2 \leq 1.5 \cdot 2$

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

Simplify $\frac{1 - 3 y}{2} \cdot 2$ .

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Step 2.2.1.1.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.1.1

Cancel the common factor.

$\frac{1 - 3 y}{2} \cdot 2 \leq 1.5 \cdot 2$

Step 2.2.1.1.1.2

Rewrite the expression.

$1 - 3 y \leq 1.5 \cdot 2$

Step 2.2.1.1.2

Reorder $1$ and $- 3 y$ .

$- 3 y + 1 \leq 1.5 \cdot 2$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $1.5$ by $2$ .

$- 3 y + 1 \leq 3$

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Subtract $1$ from both sides of the inequality.

$- 3 y \leq 3 - 1$

Step 2.3.1.2

Subtract $1$ from $3$ .

$- 3 y \leq 2$

Step 2.3.2

Divide each term in $- 3 y \leq 2$ by $- 3$ and simplify.

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Step 2.3.2.1

Divide each term in $- 3 y \leq 2$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 y}{- 3} \geq \frac{2}{- 3}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} \geq \frac{2}{- 3}$

Step 2.3.2.2.1.2

Divide $y$ by $1$ .

$y \geq \frac{2}{- 3}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Cancel the common factor of $2$ and $- 3$ .

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Step 2.3.2.3.1.1

Rewrite $2$ as $1 (2)$ .

$y \geq \frac{1 (2)}{- 3}$

Step 2.3.2.3.1.2

Cancel the common factors.

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Step 2.3.2.3.1.2.1

Rewrite $- 3$ as $1 (- 3)$ .

$y \geq \frac{1 \cdot 2}{1 \cdot - 3}$

Step 2.3.2.3.1.2.2

Cancel the common factor.

$y \geq \frac{1 \cdot 2}{1 \cdot - 3}$

Step 2.3.2.3.1.2.3

Rewrite the expression.

$y \geq \frac{2}{- 3}$

Step 2.3.2.3.2

Move the negative in front of the fraction.

$y \geq - \frac{2}{3}$

Step 3

Find the intersection of $y \leq 2$ and $y \geq - \frac{2}{3}$ .

$- \frac{2}{3} \leq y \leq 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{2}{3} \leq y \leq 2$

Interval Notation:

$[- \frac{2}{3}, 2]$

Step 5