Solve the Inequality for x 4^(2x+7)<=32^(2x-3)

$4^{2 x + 7} \leq 32^{2 x - 3}$

Step 1

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

$2^{2 (2 x + 7)} \leq 2^{5 (2 x - 3)}$

Step 2

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

$2 (2 x + 7) \leq 5 (2 x - 3)$

Step 3

Solve for $x$ .

Simplify $2 (2 x + 7)$ .

Start simplifying.

$0 + 0 + 2 (2 x + 7) \leq 5 (2 x - 3)$

Simplify by adding zeros.

$2 (2 x + 7) \leq 5 (2 x - 3)$

Apply the distributive property.

$2 (2 x) + 2 \cdot 7 \leq 5 (2 x - 3)$

Multiply.

Multiply $2$ by $2$ .

$4 x + 2 \cdot 7 \leq 5 (2 x - 3)$

Multiply $2$ by $7$ .

$4 x + 14 \leq 5 (2 x - 3)$

Simplify $5 (2 x - 3)$ .

Apply the distributive property.

$4 x + 14 \leq 5 (2 x) + 5 \cdot - 3$

Multiply.

Multiply $2$ by $5$ .

$4 x + 14 \leq 10 x + 5 \cdot - 3$

Multiply $5$ by $- 3$ .

$4 x + 14 \leq 10 x - 15$

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

Subtract $10 x$ from both sides of the inequality.

$4 x + 14 - 10 x \leq - 15$

Subtract $10 x$ from $4 x$ .

$- 6 x + 14 \leq - 15$

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

Subtract $14$ from both sides of the inequality.

$- 6 x \leq - 15 - 14$

Subtract $14$ from $- 15$ .

$- 6 x \leq - 29$

Divide each term in $- 6 x \leq - 29$ by $- 6$ and simplify.

Divide each term in $- 6 x \leq - 29$ by $- 6$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 6 x}{- 6} \geq \frac{- 29}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} \geq \frac{- 29}{- 6}$

Divide $x$ by $1$ .

$x \geq \frac{- 29}{- 6}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \geq \frac{29}{6}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq \frac{29}{6}$

Interval Notation:

$[\frac{29}{6}, \infty)$

Step 5

Solve the Inequality for x 4^(2x+7)<=32^(2x-3)

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