Trigonometry Examples

${(\frac{1}{3})}^{x} - 9 < 18$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{x}}{3^{x}} - 9 < 18$

Step 1.2

One to any power is one.

$\frac{1}{3^{x}} - 9 < 18$

Step 2

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

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Step 2.1

Add $9$ to both sides of the inequality.

$\frac{1}{3^{x}} < 18 + 9$

Step 2.2

Add $18$ and $9$ .

$\frac{1}{3^{x}} < 27$

Step 3

Move $3^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{- x} < 27$

Step 4

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

$3^{- x} < 3^{3}$

Step 5

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

$- x < 3$

Step 6

Divide each term in $- x < 3$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x < 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{3}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{3}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x > \frac{3}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $3$ by $- 1$ .

$x > - 3$

Step 7

Convert the inequality to interval notation.

$(- 3, \infty)$

Step 8