Algebra Examples

$\frac{2 x - 1}{3} < 1 - \frac{x - 2}{5}$

Step 1

Multiply both sides by $3$ .

$\frac{2 x - 1}{3} \cdot 3 < (1 - \frac{x - 2}{5}) \cdot 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $3$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{2 x - 1}{3} \cdot 3 < (1 - \frac{x - 2}{5}) \cdot 3$

Step 2.1.1.2

Rewrite the expression.

$2 x - 1 < (1 - \frac{x - 2}{5}) \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(1 - \frac{x - 2}{5}) \cdot 3$ .

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Step 2.2.1.1

Combine into one fraction.

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Step 2.2.1.1.1

Write $1$ as a fraction with a common denominator.

$2 x - 1 < (\frac{5}{5} - \frac{x - 2}{5}) \cdot 3$

Step 2.2.1.1.2

Combine the numerators over the common denominator.

$2 x - 1 < \frac{5 - (x - 2)}{5} \cdot 3$

Step 2.2.1.2

Simplify the numerator.

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Step 2.2.1.2.1

Apply the distributive property.

$2 x - 1 < \frac{5 - x - - 2}{5} \cdot 3$

Step 2.2.1.2.2

Multiply $- 1$ by $- 2$ .

$2 x - 1 < \frac{5 - x + 2}{5} \cdot 3$

Step 2.2.1.2.3

Add $5$ and $2$ .

$2 x - 1 < \frac{- x + 7}{5} \cdot 3$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $\frac{- x + 7}{5}$ and $3$ .

$2 x - 1 < \frac{(- x + 7) \cdot 3}{5}$

Step 2.2.1.3.2

Move $3$ to the left of $- x + 7$ .

$2 x - 1 < \frac{3 \cdot (- x + 7)}{5}$

Step 2.2.1.3.3

Factor $- 1$ out of $- x$ .

$2 x - 1 < \frac{3 (- (x) + 7)}{5}$

Step 2.2.1.3.4

Rewrite $7$ as $- 1 (- 7)$ .

$2 x - 1 < \frac{3 (- (x) - 1 (- 7))}{5}$

Step 2.2.1.3.5

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$2 x - 1 < \frac{3 (- (x - 7))}{5}$

Step 2.2.1.3.6

Simplify the expression.

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Step 2.2.1.3.6.1

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$2 x - 1 < \frac{3 (- 1 (x - 7))}{5}$

Step 2.2.1.3.6.2

Move the negative in front of the fraction.

$2 x - 1 < - \frac{3 (x - 7)}{5}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $5$ .

$(2 x - 1) \cdot 5 < - \frac{3 (x - 7)}{5} \cdot 5$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(2 x - 1) \cdot 5$ .

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Step 3.2.1.1.1

Apply the distributive property.

$2 x \cdot 5 - 1 \cdot 5 < - \frac{3 (x - 7)}{5} \cdot 5$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $5$ by $2$ .

$10 x - 1 \cdot 5 < - \frac{3 (x - 7)}{5} \cdot 5$

Step 3.2.1.1.2.2

Multiply $- 1$ by $5$ .

$10 x - 5 < - \frac{3 (x - 7)}{5} \cdot 5$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{3 (x - 7)}{5} \cdot 5$ .

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Step 3.2.2.1.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1.1

Move the leading negative in $- \frac{3 (x - 7)}{5}$ into the numerator.

$10 x - 5 < \frac{- 3 (x - 7)}{5} \cdot 5$

Step 3.2.2.1.1.2

Cancel the common factor.

$10 x - 5 < \frac{- 3 (x - 7)}{5} \cdot 5$

Step 3.2.2.1.1.3

Rewrite the expression.

$10 x - 5 < - 3 (x - 7)$

Step 3.2.2.1.2

Apply the distributive property.

$10 x - 5 < - 3 x - 3 \cdot - 7$

Step 3.2.2.1.3

Multiply $- 3$ by $- 7$ .

$10 x - 5 < - 3 x + 21$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Add $3 x$ to both sides of the inequality.

$10 x - 5 + 3 x < 21$

Step 3.3.1.2

Add $10 x$ and $3 x$ .

$13 x - 5 < 21$

Step 3.3.2

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

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Step 3.3.2.1

Add $5$ to both sides of the inequality.

$13 x < 21 + 5$

Step 3.3.2.2

Add $21$ and $5$ .

$13 x < 26$

Step 3.3.3

Divide each term in $13 x < 26$ by $13$ and simplify.

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Step 3.3.3.1

Divide each term in $13 x < 26$ by $13$ .

$\frac{13 x}{13} < \frac{26}{13}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $13$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{13 x}{13} < \frac{26}{13}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{26}{13}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $26$ by $13$ .

$x < 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < 2$

Interval Notation:

$(- \infty, 2)$