Question

3^(2*(2x+5))*5^(2*(3x+1))-15^(5x+6)<0

Answer 1

$3^{2(2x+5)}\cdot 5^{2(3x+1)}-15^{5x+6}$

Answer 2

To solve the inequality $3^{2(2x+5)} \cdot 5^{2(3x+1)} - 15^{5x+6} < 0$, we'll need to use some properties of exponents and basic algebraic manipulation.

First, let's rewrite the expressions:

$3^{2(2x+5)} = 9^{2x+5}$

$5^{2(3x+1)} = 25^{3x+1}$

$15^{5x+6}$

Now, let's substitute these back into the inequality:

$9^{2x+5} \cdot 25^{3x+1} - 15^{5x+6} < 0$

Next, let's try to get rid of the bases by using a common base. Since $9 = 3^2$ and $25 = 5^2$, we can rewrite them:

$(3^2)^{2x+5} \cdot (5^2)^{3x+1} - (3 \cdot 5)^{2(5x+6)} < 0$

$3^{4x+10} \cdot 5^{6x+2} - 15^{10x+12} < 0$

Now we have a single base. We can substitute $a = 3^{4x+10}$, $b = 5^{6x+2}$, and $c = 15^{10x+12}$:

$a \cdot b - c < 0$

Now, let's solve for $a$, $b$, and $c$ in terms of $x$:

$a = 3^{4x+10}$

$b = 5^{6x+2}$

$c = 15^{10x+12} = (3 \cdot 5)^{10x+12} = 3^{10x+12} \cdot 5^{10x+12} = 3^{10x+12} \cdot (5^{6x+2})^{\frac{5}{3}} = 3^{10x+12} \cdot b^{\frac{5}{3}}$

Now, let's rewrite the inequality with $a$, $b$, and $c$:

$ab - c < 0$

$(3^{4x+10})(5^{6x+2}) - (3^{10x+12})(5^{(6x+2)\cdot\frac{5}{3}}) < 0$

$3^{4x+10} \cdot 5^{6x+2} - 3^{10x+12} \cdot 5^{4x+3} < 0$

Finally, we have an inequality in terms of $x$:

$3^{4x+10} \cdot 5^{6x+2} - 3^{10x+12} \cdot 5^{4x+3} < 0$

Please note that this is a complex equation and may not have a straightforward solution. You may need to use numerical methods or a graphing calculator to find an approximate solution.