Algebra Examples

$7^{3 x} < 49^{1 - x}$

Step 1

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

$7^{3 x} < 7^{2 (1 - x)}$

Step 2

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

$3 x < 2 (1 - x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (1 - x)$ .

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Step 3.1.1

Apply the distributive property.

$3 x < 2 \cdot 1 + 2 (- x)$

Step 3.1.2

Multiply.

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Step 3.1.2.1

Multiply $2$ by $1$ .

$3 x < 2 + 2 (- x)$

Step 3.1.2.2

Multiply $- 1$ by $2$ .

$3 x < 2 - 2 x$

Step 3.2

Move all terms containing $x$ to the left side of the inequality.

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Step 3.2.1

Add $2 x$ to both sides of the inequality.

$3 x + 2 x < 2$

Step 3.2.2

Add $3 x$ and $2 x$ .

$5 x < 2$

Step 3.3

Divide each term in $5 x < 2$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 x < 2$ by $5$ .

$\frac{5 x}{5} < \frac{2}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} < \frac{2}{5}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{2}{5}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{2}{5}$

Interval Notation:

$(- \infty, \frac{2}{5})$

Step 5