Solve 12-(2^15x-2^13):(2^4)^3geq3x-2(3-2x)

$$12 - ( 2 ^ { 15 } x - 2 ^ { 13 } ) : ( 2 ^ { 4 } ) ^ { 3 } \geq 3 x - 2 ( 3 - 2 x )$$

Solve for x

$$x\leq \frac{4}{3}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$12-\frac{2^{15}x-2^{13}}{2^{12}}\geq 3x-2\left(3-2x\right)$$

Calculate 2 to the power of 15 and get 32768.

$$12-\frac{32768x-2^{13}}{2^{12}}\geq 3x-2\left(3-2x\right)$$

Calculate 2 to the power of 13 and get 8192.

$$12-\frac{32768x-8192}{2^{12}}\geq 3x-2\left(3-2x\right)$$

Calculate 2 to the power of 12 and get 4096.

$$12-\frac{32768x-8192}{4096}\geq 3x-2\left(3-2x\right)$$

Use the distributive property to multiply -2 by 3-2x.

$$12-\frac{32768x-8192}{4096}\geq 3x-6+4x$$

Combine 3x and 4x to get 7x.

$$12-\frac{32768x-8192}{4096}\geq 7x-6$$

Divide each term of 32768x-8192 by 4096 to get 8x-2.

$$12-\left(8x-2\right)\geq 7x-6$$

To find the opposite of 8x-2, find the opposite of each term.

$$12-8x+2\geq 7x-6$$

Add 12 and 2 to get 14.

$$14-8x\geq 7x-6$$

Subtract 7x from both sides.

$$14-8x-7x\geq -6$$

Combine -8x and -7x to get -15x.

$$14-15x\geq -6$$

Subtract 14 from both sides.

$$-15x\geq -6-14$$

Subtract 14 from -6 to get -20.

$$-15x\geq -20$$

Divide both sides by -15. Since -15 is negative, the inequality direction is changed.

$$x\leq \frac{-20}{-15}$$

Reduce the fraction \frac{-20}{-15} to lowest terms by extracting and canceling out -5.

$$x\leq \frac{4}{3}$$