Algebra Examples

$\frac{1}{4} (\frac{5}{12} x - \frac{3}{4}) > x + \frac{1}{12}$

Step 1

Simplify $\frac{1}{4} \cdot (\frac{5}{12} x - \frac{3}{4})$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{4} \cdot (\frac{5}{12} x - \frac{3}{4}) > x + \frac{1}{12}$

Step 1.2

Simplify by adding zeros.

$\frac{1}{4} \cdot (\frac{5}{12} x - \frac{3}{4}) > x + \frac{1}{12}$

Step 1.3

Combine $\frac{5}{12}$ and $x$ .

$\frac{1}{4} \cdot (\frac{5 x}{12} - \frac{3}{4}) > x + \frac{1}{12}$

Step 1.4

Apply the distributive property.

$\frac{1}{4} \cdot \frac{5 x}{12} + \frac{1}{4} (- \frac{3}{4}) > x + \frac{1}{12}$

Step 1.5

Multiply $\frac{1}{4} \cdot \frac{5 x}{12}$ .

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Step 1.5.1

Multiply $\frac{1}{4}$ by $\frac{5 x}{12}$ .

$\frac{5 x}{4 \cdot 12} + \frac{1}{4} (- \frac{3}{4}) > x + \frac{1}{12}$

Step 1.5.2

Multiply $4$ by $12$ .

$\frac{5 x}{48} + \frac{1}{4} (- \frac{3}{4}) > x + \frac{1}{12}$

Step 1.6

Multiply $\frac{1}{4} (- \frac{3}{4})$ .

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Step 1.6.1

Multiply $\frac{1}{4}$ by $\frac{3}{4}$ .

$\frac{5 x}{48} - \frac{3}{4 \cdot 4} > x + \frac{1}{12}$

Step 1.6.2

Multiply $4$ by $4$ .

$\frac{5 x}{48} - \frac{3}{16} > x + \frac{1}{12}$

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.

$\frac{5 x}{48} - \frac{3}{16} - x > \frac{1}{12}$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{48}{48}$ .

$\frac{5 x}{48} - x \cdot \frac{48}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.3

Combine $- x$ and $\frac{48}{48}$ .

$\frac{5 x}{48} + \frac{- x \cdot 48}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 x - x \cdot 48}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $x$ out of $5 x - x \cdot 48$ .

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Step 2.5.1.1.1

Factor $x$ out of $5 x$ .

$\frac{x \cdot 5 - x \cdot 48}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.1.1.2

Factor $x$ out of $- x \cdot 48$ .

$\frac{x \cdot 5 + x (- 1 \cdot 48)}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.1.1.3

Factor $x$ out of $x \cdot 5 + x (- 1 \cdot 48)$ .

$\frac{x (5 - 1 \cdot 48)}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.1.2

Multiply $- 1$ by $48$ .

$\frac{x (5 - 48)}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.1.3

Subtract $48$ from $5$ .

$\frac{x \cdot - 43}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.2

Move $- 43$ to the left of $x$ .

$\frac{- 43 \cdot x}{48} - \frac{3}{16} > \frac{1}{12}$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{43 x}{48} - \frac{3}{16} > \frac{1}{12}$

Step 3

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

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Step 3.1

Add $\frac{3}{16}$ to both sides of the inequality.

$- \frac{43 x}{48} > \frac{1}{12} + \frac{3}{16}$

Step 3.2

To write $\frac{1}{12}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{43 x}{48} > \frac{1}{12} \cdot \frac{4}{4} + \frac{3}{16}$

Step 3.3

To write $\frac{3}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{43 x}{48} > \frac{1}{12} \cdot \frac{4}{4} + \frac{3}{16} \cdot \frac{3}{3}$

Step 3.4

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{1}{12}$ by $\frac{4}{4}$ .

$- \frac{43 x}{48} > \frac{4}{12 \cdot 4} + \frac{3}{16} \cdot \frac{3}{3}$

Step 3.4.2

Multiply $12$ by $4$ .

$- \frac{43 x}{48} > \frac{4}{48} + \frac{3}{16} \cdot \frac{3}{3}$

Step 3.4.3

Multiply $\frac{3}{16}$ by $\frac{3}{3}$ .

$- \frac{43 x}{48} > \frac{4}{48} + \frac{3 \cdot 3}{16 \cdot 3}$

Step 3.4.4

Multiply $16$ by $3$ .

$- \frac{43 x}{48} > \frac{4}{48} + \frac{3 \cdot 3}{48}$

Step 3.5

Combine the numerators over the common denominator.

$- \frac{43 x}{48} > \frac{4 + 3 \cdot 3}{48}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $3$ .

$- \frac{43 x}{48} > \frac{4 + 9}{48}$

Step 3.6.2

Add $4$ and $9$ .

$- \frac{43 x}{48} > \frac{13}{48}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 43 x > 13$

Step 5

Divide each term in $- 43 x > 13$ by $- 43$ and simplify.

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Step 5.1

Divide each term in $- 43 x > 13$ by $- 43$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 43 x}{- 43} < \frac{13}{- 43}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 43$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 43 x}{- 43} < \frac{13}{- 43}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x < \frac{13}{- 43}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x < - \frac{13}{43}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{13}{43}$

Interval Notation:

$(- \infty, - \frac{13}{43})$