Algebra Examples

$\frac{- 10 + x}{4} + 5 \geq \frac{7 x - 5}{3}$

Step 1

Multiply both sides by $3$ .

$(\frac{- 10 + x}{4} + 5) \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{- 10 + x}{4} + 5) \cdot 3$ .

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Step 2.1.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(\frac{- 10 + x}{4} + 5 \cdot \frac{4}{4}) \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.2

Simplify terms.

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Step 2.1.1.2.1

Combine $5$ and $\frac{4}{4}$ .

$(\frac{- 10 + x}{4} + \frac{5 \cdot 4}{4}) \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.2.2

Combine the numerators over the common denominator.

$\frac{- 10 + x + 5 \cdot 4}{4} \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.3

Simplify the numerator.

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Step 2.1.1.3.1

Multiply $5$ by $4$ .

$\frac{- 10 + x + 20}{4} \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.3.2

Add $- 10$ and $20$ .

$\frac{x + 10}{4} \cdot 3 \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.4

Combine fractions.

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Step 2.1.1.4.1

Combine $\frac{x + 10}{4}$ and $3$ .

$\frac{(x + 10) \cdot 3}{4} \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.1.1.4.2

Move $3$ to the left of $x + 10$ .

$\frac{3 (x + 10)}{4} \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 (x + 10)}{4} \geq \frac{7 x - 5}{3} \cdot 3$

Step 2.2.1.2

Rewrite the expression.

$\frac{3 (x + 10)}{4} \geq 7 x - 5$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{3 (x + 10)}{4} \cdot 4 \geq (7 x - 5) \cdot 4$

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 $\frac{3 (x + 10)}{4} \cdot 4$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{3 (x + 10)}{4} \cdot 4 \geq (7 x - 5) \cdot 4$

Step 3.2.1.1.1.2

Rewrite the expression.

$3 (x + 10) \geq (7 x - 5) \cdot 4$

Step 3.2.1.1.2

Apply the distributive property.

$3 x + 3 \cdot 10 \geq (7 x - 5) \cdot 4$

Step 3.2.1.1.3

Multiply $3$ by $10$ .

$3 x + 30 \geq (7 x - 5) \cdot 4$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(7 x - 5) \cdot 4$ .

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Step 3.2.2.1.1

Apply the distributive property.

$3 x + 30 \geq 7 x \cdot 4 - 5 \cdot 4$

Step 3.2.2.1.2

Multiply.

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Step 3.2.2.1.2.1

Multiply $4$ by $7$ .

$3 x + 30 \geq 28 x - 5 \cdot 4$

Step 3.2.2.1.2.2

Multiply $- 5$ by $4$ .

$3 x + 30 \geq 28 x - 20$

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

Subtract $28 x$ from both sides of the inequality.

$3 x + 30 - 28 x \geq - 20$

Step 3.3.1.2

Subtract $28 x$ from $3 x$ .

$- 25 x + 30 \geq - 20$

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

Subtract $30$ from both sides of the inequality.

$- 25 x \geq - 20 - 30$

Step 3.3.2.2

Subtract $30$ from $- 20$ .

$- 25 x \geq - 50$

Step 3.3.3

Divide each term in $- 25 x \geq - 50$ by $- 25$ and simplify.

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Step 3.3.3.1

Divide each term in $- 25 x \geq - 50$ by $- 25$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 25 x}{- 25} \leq \frac{- 50}{- 25}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 25$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 25 x}{- 25} \leq \frac{- 50}{- 25}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 50}{- 25}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 50$ by $- 25$ .

$x \leq 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq 2$

Interval Notation:

$(- \infty, 2]$