Algebra Examples

$3 {(\frac{1}{4})}^{x + 1} < 192$

Step 1

Divide each term in $3 {(\frac{1}{4})}^{x + 1} < 192$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 {(\frac{1}{4})}^{x + 1} < 192$ by $3$ .

$\frac{3 {(\frac{1}{4})}^{x + 1}}{3} < \frac{192}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 {(\frac{1}{4})}^{x + 1}}{3} < \frac{192}{3}$

Step 1.2.1.2

Divide ${(\frac{1}{4})}^{x + 1}$ by $1$ .

${(\frac{1}{4})}^{x + 1} < \frac{192}{3}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{4}$ .

$\frac{1^{x + 1}}{4^{x + 1}} < \frac{192}{3}$

Step 1.2.2.2

One to any power is one.

$\frac{1}{4^{x + 1}} < \frac{192}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $192$ by $3$ .

$\frac{1}{4^{x + 1}} < 64$

Step 2

Move $4^{x + 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- (x + 1)} < 64$

Step 3

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

$4^{- (x + 1)} < 4^{3}$

Step 4

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

$- (x + 1) < 3$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $- (x + 1)$ .

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Step 5.1.1

Apply the distributive property.

$- x - 1 \cdot 1 < 3$

Step 5.1.2

Multiply $- 1$ by $1$ .

$- x - 1 < 3$

Step 5.2

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

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Step 5.2.1

Add $1$ to both sides of the inequality.

$- x < 3 + 1$

Step 5.2.2

Add $3$ and $1$ .

$- x < 4$

Step 5.3

Divide each term in $- x < 4$ by $- 1$ and simplify.

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Step 5.3.1

Divide each term in $- x < 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{4}{- 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{4}{- 1}$

Step 5.3.2.2

Divide $x$ by $1$ .

$x > \frac{4}{- 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $4$ by $- 1$ .

$x > - 4$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > - 4$

Interval Notation:

$(- 4, \infty)$

Step 7