Solve for x 16^(2x-5)>=64^(2x-3)

$16^{2 x - 5} \geq 64^{2 x - 3}$

Step 1

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

$4^{2 (2 x - 5)} \geq 4^{3 (2 x - 3)}$

Step 2

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

$2 (2 x - 5) \geq 3 (2 x - 3)$

Step 3

Solve for $x$ .

Simplify $2 (2 x - 5)$ .

Start simplifying.

$0 + 0 + 2 (2 x - 5) \geq 3 (2 x - 3)$

Simplify by adding zeros.

$2 (2 x - 5) \geq 3 (2 x - 3)$

Apply the distributive property.

$2 (2 x) + 2 \cdot - 5 \geq 3 (2 x - 3)$

Multiply.

Multiply $2$ by $2$ .

$4 x + 2 \cdot - 5 \geq 3 (2 x - 3)$

Multiply $2$ by $- 5$ .

$4 x - 10 \geq 3 (2 x - 3)$

Simplify $3 (2 x - 3)$ .

Apply the distributive property.

$4 x - 10 \geq 3 (2 x) + 3 \cdot - 3$

Multiply.

Multiply $2$ by $3$ .

$4 x - 10 \geq 6 x + 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$4 x - 10 \geq 6 x - 9$

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

Subtract $6 x$ from both sides of the inequality.

$4 x - 10 - 6 x \geq - 9$

Subtract $6 x$ from $4 x$ .

$- 2 x - 10 \geq - 9$

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

Add $10$ to both sides of the inequality.

$- 2 x \geq - 9 + 10$

Add $- 9$ and $10$ .

$- 2 x \geq 1$

Divide each term in $- 2 x \geq 1$ by $- 2$ and simplify.

Divide each term in $- 2 x \geq 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{1}{- 2}$

Divide $x$ by $1$ .

$x \leq \frac{1}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{1}{2}$

Interval Notation:

$(- \infty, - \frac{1}{2}]$

Step 5

Solve for x 16^(2x-5)>=64^(2x-3)

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