Algebra Examples

$\frac{5 x - 1}{5} + \frac{x + 1}{2} \leq x$

Step 1

Simplify $\frac{5 x - 1}{5} + \frac{x + 1}{2}$ .

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Step 1.1

To write $\frac{5 x - 1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2

To write $\frac{x + 1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{5 x - 1}{5}$ by $\frac{2}{2}$ .

$\frac{(5 x - 1) \cdot 2}{5 \cdot 2} + \frac{x + 1}{2} \cdot \frac{5}{5} \leq x$

Step 1.3.2

Multiply $5$ by $2$ .

$\frac{(5 x - 1) \cdot 2}{10} + \frac{x + 1}{2} \cdot \frac{5}{5} \leq x$

Step 1.3.3

Multiply $\frac{x + 1}{2}$ by $\frac{5}{5}$ .

$\frac{(5 x - 1) \cdot 2}{10} + \frac{(x + 1) \cdot 5}{2 \cdot 5} \leq x$

Step 1.3.4

Multiply $2$ by $5$ .

$\frac{(5 x - 1) \cdot 2}{10} + \frac{(x + 1) \cdot 5}{10} \leq x$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(5 x - 1) \cdot 2 + (x + 1) \cdot 5}{10} \leq x$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{5 x \cdot 2 - 1 \cdot 2 + (x + 1) \cdot 5}{10} \leq x$

Step 1.5.2

Multiply $2$ by $5$ .

$\frac{10 x - 1 \cdot 2 + (x + 1) \cdot 5}{10} \leq x$

Step 1.5.3

Multiply $- 1$ by $2$ .

$\frac{10 x - 2 + (x + 1) \cdot 5}{10} \leq x$

Step 1.5.4

Apply the distributive property.

$\frac{10 x - 2 + x \cdot 5 + 1 \cdot 5}{10} \leq x$

Step 1.5.5

Move $5$ to the left of $x$ .

$\frac{10 x - 2 + 5 \cdot x + 1 \cdot 5}{10} \leq x$

Step 1.5.6

Multiply $5$ by $1$ .

$\frac{10 x - 2 + 5 \cdot x + 5}{10} \leq x$

Step 1.5.7

Add $10 x$ and $5 x$ .

$\frac{15 x - 2 + 5}{10} \leq x$

Step 1.5.8

Add $- 2$ and $5$ .

$\frac{15 x + 3}{10} \leq x$

Step 1.5.9

Factor $3$ out of $15 x + 3$ .

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Step 1.5.9.1

Factor $3$ out of $15 x$ .

$\frac{3 (5 x) + 3}{10} \leq x$

Step 1.5.9.2

Factor $3$ out of $3$ .

$\frac{3 (5 x) + 3 (1)}{10} \leq x$

Step 1.5.9.3

Factor $3$ out of $3 (5 x) + 3 (1)$ .

$\frac{3 (5 x + 1)}{10} \leq x$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the inequality.

$\frac{3 (5 x + 1)}{10} - x \leq 0$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{3 (5 x + 1)}{10} - x \cdot \frac{10}{10} \leq 0$

Step 2.3

Combine $- x$ and $\frac{10}{10}$ .

$\frac{3 (5 x + 1)}{10} + \frac{- x \cdot 10}{10} \leq 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 (5 x + 1) - x \cdot 10}{10} \leq 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{3 (5 x) + 3 \cdot 1 - x \cdot 10}{10} \leq 0$

Step 2.5.2

Multiply $5$ by $3$ .

$\frac{15 x + 3 \cdot 1 - x \cdot 10}{10} \leq 0$

Step 2.5.3

Multiply $3$ by $1$ .

$\frac{15 x + 3 - x \cdot 10}{10} \leq 0$

Step 2.5.4

Multiply $10$ by $- 1$ .

$\frac{15 x + 3 - 10 x}{10} \leq 0$

Step 2.5.5

Subtract $10 x$ from $15 x$ .

$\frac{5 x + 3}{10} \leq 0$

Step 3

Multiply both sides by $10$ .

$\frac{5 x + 3}{10} \cdot 10 \leq 0 \cdot 10$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $10$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{5 x + 3}{10} \cdot 10 \leq 0 \cdot 10$

Step 4.1.1.2

Rewrite the expression.

$5 x + 3 \leq 0 \cdot 10$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $0$ by $10$ .

$5 x + 3 \leq 0$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $3$ from both sides of the inequality.

$5 x \leq - 3$

Step 5.2

Divide each term in $5 x \leq - 3$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 x \leq - 3$ by $5$ .

$\frac{5 x}{5} \leq \frac{- 3}{5}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{- 3}{5}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 3}{5}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{3}{5}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{3}{5}$

Interval Notation:

$(- \infty, - \frac{3}{5}]$