Algebra Examples

$- 4 < \frac{y - 7}{- 3} \leq 2$

Step 1

Solve $- 4 < \frac{y - 7}{- 3}$ for $y$ .

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Step 1.1

Multiply both sides by $- 3$ .

$- 4 \cdot - 3 > \frac{y - 7}{- 3} \cdot - 3$

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 $- 4$ by $- 3$ .

$12 > \frac{y - 7}{- 3} \cdot - 3$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Simplify $\frac{y - 7}{- 3} \cdot - 3$ .

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Step 1.2.2.1.1

Simplify terms.

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Step 1.2.2.1.1.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1.1.1

Factor $3$ out of $- 3$ .

$12 > \frac{y - 7}{3 (- 1)} \cdot - 3$

Step 1.2.2.1.1.1.2

Factor $3$ out of $- 3$ .

$12 > \frac{y - 7}{3 \cdot - 1} \cdot (3 \cdot - 1)$

Step 1.2.2.1.1.1.3

Cancel the common factor.

$12 > \frac{y - 7}{3 \cdot - 1} \cdot (3 \cdot - 1)$

Step 1.2.2.1.1.1.4

Rewrite the expression.

$12 > \frac{y - 7}{- 1} \cdot - 1$

Step 1.2.2.1.1.2

Combine $\frac{y - 7}{- 1}$ and $- 1$ .

$12 > \frac{(y - 7) \cdot - 1}{- 1}$

Step 1.2.2.1.1.3

Simplify the expression.

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Step 1.2.2.1.1.3.1

Move the negative one from the denominator of $\frac{(y - 7) \cdot - 1}{- 1}$ .

$12 > - 1 \cdot ((y - 7) \cdot - 1)$

Step 1.2.2.1.1.3.2

Rewrite $- 1 \cdot ((y - 7) \cdot - 1)$ as $- ((y - 7) \cdot - 1)$ .

$12 > - ((y - 7) \cdot - 1)$

Step 1.2.2.1.1.4

Apply the distributive property.

$12 > - (y \cdot - 1 - 7 \cdot - 1)$

Step 1.2.2.1.1.5

Simplify the expression.

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Step 1.2.2.1.1.5.1

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

$12 > - (- 1 \cdot y - 7 \cdot - 1)$

Step 1.2.2.1.1.5.2

Multiply $- 7$ by $- 1$ .

$12 > - (- 1 \cdot y + 7)$

Step 1.2.2.1.2

Rewrite $- 1 y$ as $- y$ .

$12 > - (- y + 7)$

Step 1.2.2.1.3

Apply the distributive property.

$12 > - - y - 1 \cdot 7$

Step 1.2.2.1.4

Multiply $- - y$ .

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Step 1.2.2.1.4.1

Multiply $- 1$ by $- 1$ .

$12 > 1 y - 1 \cdot 7$

Step 1.2.2.1.4.2

Multiply $y$ by $1$ .

$12 > y - 1 \cdot 7$

Step 1.2.2.1.5

Multiply $- 1$ by $7$ .

$12 > y - 7$

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.

$y - 7 < 12$

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

Add $7$ to both sides of the inequality.

$y < 12 + 7$

Step 1.3.2.2

Add $12$ and $7$ .

$y < 19$

Step 2

Solve $\frac{y - 7}{- 3} \leq 2$ for $y$ .

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Step 2.1

Multiply both sides by $- 3$ .

$\frac{y - 7}{- 3} \cdot - 3 \geq 2 \cdot - 3$

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

Simplify $\frac{y - 7}{- 3} \cdot - 3$ .

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Step 2.2.1.1.1

Simplify terms.

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Step 2.2.1.1.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1.1.1

Factor $3$ out of $- 3$ .

$\frac{y - 7}{3 (- 1)} \cdot - 3 \geq 2 \cdot - 3$

Step 2.2.1.1.1.1.2

Factor $3$ out of $- 3$ .

$\frac{y - 7}{3 \cdot - 1} \cdot (3 \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.1.3

Cancel the common factor.

$\frac{y - 7}{3 \cdot - 1} \cdot (3 \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.1.4

Rewrite the expression.

$\frac{y - 7}{- 1} \cdot - 1 \geq 2 \cdot - 3$

Step 2.2.1.1.1.2

Combine $\frac{y - 7}{- 1}$ and $- 1$ .

$\frac{(y - 7) \cdot - 1}{- 1} \geq 2 \cdot - 3$

Step 2.2.1.1.1.3

Simplify the expression.

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Step 2.2.1.1.1.3.1

Move the negative one from the denominator of $\frac{(y - 7) \cdot - 1}{- 1}$ .

$- 1 \cdot ((y - 7) \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.3.2

Rewrite $- 1 \cdot ((y - 7) \cdot - 1)$ as $- ((y - 7) \cdot - 1)$ .

$- ((y - 7) \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.4

Apply the distributive property.

$- (y \cdot - 1 - 7 \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.5

Simplify the expression.

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Step 2.2.1.1.1.5.1

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

$- (- 1 \cdot y - 7 \cdot - 1) \geq 2 \cdot - 3$

Step 2.2.1.1.1.5.2

Multiply $- 7$ by $- 1$ .

$- (- 1 \cdot y + 7) \geq 2 \cdot - 3$

Step 2.2.1.1.2

Rewrite $- 1 y$ as $- y$ .

$- (- y + 7) \geq 2 \cdot - 3$

Step 2.2.1.1.3

Apply the distributive property.

$- - y - 1 \cdot 7 \geq 2 \cdot - 3$

Step 2.2.1.1.4

Multiply $- - y$ .

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Step 2.2.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 y - 1 \cdot 7 \geq 2 \cdot - 3$

Step 2.2.1.1.4.2

Multiply $y$ by $1$ .

$y - 1 \cdot 7 \geq 2 \cdot - 3$

Step 2.2.1.1.5

Multiply $- 1$ by $7$ .

$y - 7 \geq 2 \cdot - 3$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $2$ by $- 3$ .

$y - 7 \geq - 6$

Step 2.3

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

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Step 2.3.1

Add $7$ to both sides of the inequality.

$y \geq - 6 + 7$

Step 2.3.2

Add $- 6$ and $7$ .

$y \geq 1$

Step 3

Find the intersection of $y < 19$ and $y \geq 1$ .

$1 \leq y < 19$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$1 \leq y < 19$

Interval Notation:

$[1, 19)$

Step 5