Algebra Examples

$\frac{4}{3} s - 3 < s + \frac{2}{3} - \frac{1}{3} s$

Step 1

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

$s + \frac{2}{3} - \frac{1}{3} s > \frac{4}{3} s - 3$

Step 2

Simplify $s + \frac{2}{3} - \frac{1}{3} s$ .

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Step 2.1

Combine $s$ and $\frac{1}{3}$ .

$s + \frac{2}{3} - \frac{s}{3} > \frac{4}{3} s - 3$

Step 2.2

To write $s$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$s \cdot \frac{3}{3} - \frac{s}{3} + \frac{2}{3} > \frac{4}{3} s - 3$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $s$ and $\frac{3}{3}$ .

$\frac{s \cdot 3}{3} - \frac{s}{3} + \frac{2}{3} > \frac{4}{3} s - 3$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{s \cdot 3 - s}{3} + \frac{2}{3} > \frac{4}{3} s - 3$

Step 2.3.3

Combine the numerators over the common denominator.

$\frac{s \cdot 3 - s + 2}{3} > \frac{4}{3} s - 3$

Step 2.4

Move $3$ to the left of $s$ .

$\frac{3 s - s + 2}{3} > \frac{4}{3} s - 3$

Step 2.5

Simplify terms.

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Step 2.5.1

Subtract $s$ from $3 s$ .

$\frac{2 s + 2}{3} > \frac{4}{3} s - 3$

Step 2.5.2

Factor $2$ out of $2 s + 2$ .

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Step 2.5.2.1

Factor $2$ out of $2 s$ .

$\frac{2 (s) + 2}{3} > \frac{4}{3} s - 3$

Step 2.5.2.2

Factor $2$ out of $2$ .

$\frac{2 (s) + 2 (1)}{3} > \frac{4}{3} s - 3$

Step 2.5.2.3

Factor $2$ out of $2 (s) + 2 (1)$ .

$\frac{2 (s + 1)}{3} > \frac{4}{3} s - 3$

Step 3

Combine $\frac{4}{3}$ and $s$ .

$\frac{2 (s + 1)}{3} > \frac{4 s}{3} - 3$

Step 4

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

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Step 4.1

Subtract $\frac{4 s}{3}$ from both sides of the inequality.

$\frac{2 (s + 1)}{3} - \frac{4 s}{3} > - 3$

Step 4.2

Combine the numerators over the common denominator.

$\frac{2 (s + 1) - 4 s}{3} > - 3$

Step 4.3

Simplify each term.

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Step 4.3.1

Apply the distributive property.

$\frac{2 s + 2 \cdot 1 - 4 s}{3} > - 3$

Step 4.3.2

Multiply $2$ by $1$ .

$\frac{2 s + 2 - 4 s}{3} > - 3$

Step 4.4

Subtract $4 s$ from $2 s$ .

$\frac{- 2 s + 2}{3} > - 3$

Step 4.5

Factor $2$ out of $- 2 s + 2$ .

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Step 4.5.1

Factor $2$ out of $- 2 s$ .

$\frac{2 (- s) + 2}{3} > - 3$

Step 4.5.2

Factor $2$ out of $2$ .

$\frac{2 (- s) + 2 (1)}{3} > - 3$

Step 4.5.3

Factor $2$ out of $2 (- s) + 2 (1)$ .

$\frac{2 (- s + 1)}{3} > - 3$

Step 4.6

Factor $- 1$ out of $- s$ .

$\frac{2 (- (s) + 1)}{3} > - 3$

Step 4.7

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

$\frac{2 (- (s) - 1 (- 1))}{3} > - 3$

Step 4.8

Factor $- 1$ out of $- (s) - 1 (- 1)$ .

$\frac{2 (- (s - 1))}{3} > - 3$

Step 4.9

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

$\frac{2 (- 1 (s - 1))}{3} > - 3$

Step 4.10

Move the negative in front of the fraction.

$- \frac{2 (s - 1)}{3} > - 3$

Step 5

Divide each term in $- \frac{2 (s - 1)}{3} > - 3$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{2 (s - 1)}{3} > - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 (s - 1)}{3}}{- 1} < \frac{- 3}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 (s - 1)}{3}}{1} < \frac{- 3}{- 1}$

Step 5.2.2

Divide $\frac{2 (s - 1)}{3}$ by $1$ .

$\frac{2 (s - 1)}{3} < \frac{- 3}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 3$ by $- 1$ .

$\frac{2 (s - 1)}{3} < 3$

Step 6

Multiply both sides by $3$ .

$\frac{2 (s - 1)}{3} \cdot 3 < 3 \cdot 3$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{2 (s - 1)}{3} \cdot 3$ .

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Step 7.1.1.1

Cancel the common factor of $3$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{2 (s - 1)}{3} \cdot 3 < 3 \cdot 3$

Step 7.1.1.1.2

Rewrite the expression.

$2 (s - 1) < 3 \cdot 3$

Step 7.1.1.2

Apply the distributive property.

$2 s + 2 \cdot - 1 < 3 \cdot 3$

Step 7.1.1.3

Multiply $2$ by $- 1$ .

$2 s - 2 < 3 \cdot 3$

Step 7.2

Simplify the right side.

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Step 7.2.1

Multiply $3$ by $3$ .

$2 s - 2 < 9$

Step 8

Solve for $s$ .

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Step 8.1

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

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Step 8.1.1

Add $2$ to both sides of the inequality.

$2 s < 9 + 2$

Step 8.1.2

Add $9$ and $2$ .

$2 s < 11$

Step 8.2

Divide each term in $2 s < 11$ by $2$ and simplify.

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Step 8.2.1

Divide each term in $2 s < 11$ by $2$ .

$\frac{2 s}{2} < \frac{11}{2}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{2 s}{2} < \frac{11}{2}$

Step 8.2.2.1.2

Divide $s$ by $1$ .

$s < \frac{11}{2}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$s < \frac{11}{2}$

Interval Notation:

$(- \infty, \frac{11}{2})$