Algebra Examples

$3^{5 x - 1} \leq 27^{x + 3}$

Step 1

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

$3^{5 x - 1} \leq 3^{3 (x + 3)}$

Step 2

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

$5 x - 1 \leq 3 (x + 3)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (x + 3)$ .

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Step 3.1.1

Rewrite.

$5 x - 1 \leq 0 + 0 + 3 (x + 3)$

Step 3.1.2

Simplify by adding zeros.

$5 x - 1 \leq 3 (x + 3)$

Step 3.1.3

Apply the distributive property.

$5 x - 1 \leq 3 x + 3 \cdot 3$

Step 3.1.4

Multiply $3$ by $3$ .

$5 x - 1 \leq 3 x + 9$

Step 3.2

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

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Step 3.2.1

Subtract $3 x$ from both sides of the inequality.

$5 x - 1 - 3 x \leq 9$

Step 3.2.2

Subtract $3 x$ from $5 x$ .

$2 x - 1 \leq 9$

Step 3.3

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

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Step 3.3.1

Add $1$ to both sides of the inequality.

$2 x \leq 9 + 1$

Step 3.3.2

Add $9$ and $1$ .

$2 x \leq 10$

Step 3.4

Divide each term in $2 x \leq 10$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x \leq 10$ by $2$ .

$\frac{2 x}{2} \leq \frac{10}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{10}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{10}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $10$ by $2$ .

$x \leq 5$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq 5$

Interval Notation:

$(- \infty, 5]$

Step 5