Algebra Examples

$\frac{3}{10} (t + 2) - \frac{t}{20} \geq \frac{3}{5}$

Step 1

Simplify $\frac{3}{10} \cdot (t + 2) - \frac{t}{20}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{3}{10} t + \frac{3}{10} \cdot 2 - \frac{t}{20} \geq \frac{3}{5}$

Step 1.1.2

Combine $\frac{3}{10}$ and $t$ .

$\frac{3 t}{10} + \frac{3}{10} \cdot 2 - \frac{t}{20} \geq \frac{3}{5}$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $10$ .

$\frac{3 t}{10} + \frac{3}{2 (5)} \cdot 2 - \frac{t}{20} \geq \frac{3}{5}$

Step 1.1.3.2

Cancel the common factor.

$\frac{3 t}{10} + \frac{3}{2 \cdot 5} \cdot 2 - \frac{t}{20} \geq \frac{3}{5}$

Step 1.1.3.3

Rewrite the expression.

$\frac{3 t}{10} + \frac{3}{5} - \frac{t}{20} \geq \frac{3}{5}$

Step 1.2

To write $\frac{3 t}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 t}{10} \cdot \frac{2}{2} - \frac{t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.3

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{3 t}{10}$ by $\frac{2}{2}$ .

$\frac{3 t \cdot 2}{10 \cdot 2} - \frac{t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.3.2

Multiply $10$ by $2$ .

$\frac{3 t \cdot 2}{20} - \frac{t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 t \cdot 2 - t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5

Simplify each term.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $t$ out of $3 t \cdot 2 - t$ .

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Step 1.5.1.1.1

Factor $t$ out of $3 t \cdot 2$ .

$\frac{t (3 \cdot 2) - t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.1.1.2

Factor $t$ out of $- t$ .

$\frac{t (3 \cdot 2) + t \cdot - 1}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.1.1.3

Factor $t$ out of $t (3 \cdot 2) + t \cdot - 1$ .

$\frac{t (3 \cdot 2 - 1)}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.1.2

Multiply $3$ by $2$ .

$\frac{t (6 - 1)}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.1.3

Subtract $1$ from $6$ .

$\frac{t \cdot 5}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.2

Cancel the common factor of $5$ and $20$ .

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Step 1.5.2.1

Factor $5$ out of $t \cdot 5$ .

$\frac{5 \cdot t}{20} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $5$ out of $20$ .

$\frac{5 \cdot t}{5 \cdot 4} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{5 \cdot t}{5 \cdot 4} + \frac{3}{5} \geq \frac{3}{5}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{t}{4} + \frac{3}{5} \geq \frac{3}{5}$

Step 2

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

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Step 2.1

Subtract $\frac{3}{5}$ from both sides of the inequality.

$\frac{t}{4} \geq \frac{3}{5} - \frac{3}{5}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{t}{4} \geq \frac{3 - 3}{5}$

Step 2.3

Subtract $3$ from $3$ .

$\frac{t}{4} \geq \frac{0}{5}$

Step 2.4

Divide $0$ by $5$ .

$\frac{t}{4} \geq 0$

Step 3

Multiply both sides by $4$ .

$\frac{t}{4} \cdot 4 \geq 0 \cdot 4$

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 $4$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{t}{4} \cdot 4 \geq 0 \cdot 4$

Step 4.1.1.2

Rewrite the expression.

$t \geq 0 \cdot 4$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $0$ by $4$ .

$t \geq 0$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$t \geq 0$

Interval Notation:

$[0, \infty)$