Algebra Examples

$8 \leq 16^{7 x + 2}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$2^{3} \leq 2^{4 (7 x + 2)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 \leq 4 (7 x + 2)$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite so $x$ is on the left side of the inequality.

$4 (7 x + 2) \geq 3$

Step 3.2

Divide each term in $4 (7 x + 2) \geq 3$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 (7 x + 2) \geq 3$ by $4$ .

$\frac{4 (7 x + 2)}{4} \geq \frac{3}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 (7 x + 2)}{4} \geq \frac{3}{4}$

Step 3.2.2.1.2

Divide $7 x + 2$ by $1$ .

$7 x + 2 \geq \frac{3}{4}$

Step 3.3

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

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Step 3.3.1

Subtract $2$ from both sides of the inequality.

$7 x \geq \frac{3}{4} - 2$

Step 3.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 x \geq \frac{3}{4} - 2 \cdot \frac{4}{4}$

Step 3.3.3

Combine $- 2$ and $\frac{4}{4}$ .

$7 x \geq \frac{3}{4} + \frac{- 2 \cdot 4}{4}$

Step 3.3.4

Combine the numerators over the common denominator.

$7 x \geq \frac{3 - 2 \cdot 4}{4}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Multiply $- 2$ by $4$ .

$7 x \geq \frac{3 - 8}{4}$

Step 3.3.5.2

Subtract $8$ from $3$ .

$7 x \geq \frac{- 5}{4}$

Step 3.3.6

Move the negative in front of the fraction.

$7 x \geq - \frac{5}{4}$

Step 3.4

Divide each term in $7 x \geq - \frac{5}{4}$ by $7$ and simplify.

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Step 3.4.1

Divide each term in $7 x \geq - \frac{5}{4}$ by $7$ .

$\frac{7 x}{7} \geq \frac{- \frac{5}{4}}{7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{- \frac{5}{4}}{7}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- \frac{5}{4}}{7}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \geq - \frac{5}{4} \cdot \frac{1}{7}$

Step 3.4.3.2

Multiply $- \frac{5}{4} \cdot \frac{1}{7}$ .

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Step 3.4.3.2.1

Multiply $\frac{1}{7}$ by $\frac{5}{4}$ .

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

Step 3.4.3.2.2

Multiply $7$ by $4$ .

$x \geq - \frac{5}{28}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{5}{28}$

Interval Notation:

$[- \frac{5}{28}, \infty)$

Step 5