Solve x-1/4(1-1/2-1/4)^-2+(6-3x)(1/2+frac{1}{4})^-2geq-frac{x}{3}

$$\frac { x - 1 } { 4 } ( 1 - \frac { 1 } { 2 } - \frac { 1 } { 4 } ) ^ { - 2 } + ( 6 - 3 x ) ( \frac { 1 } { 2 } + \frac { 1 } { 4 } ) ^ { - 2 } \geq - \frac { x } { 3 }$$

Solve for x

$$x\leq \frac{20}{3}$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 4,3. Since 12 is positive, the inequality direction remains the same.

$$3\left(x-1\right)\left(1-\frac{1}{2}-\frac{1}{4}\right)^{-2}+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

$$3\left(x-1\right)\left(\frac{1}{2}-\frac{1}{4}\right)^{-2}+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Subtract \frac{1}{4} from \frac{1}{2} to get \frac{1}{4}.

$$3\left(x-1\right)\times \left(\frac{1}{4}\right)^{-2}+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Calculate \frac{1}{4} to the power of -2 and get 16.

$$3\left(x-1\right)\times 16+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Multiply 3 and 16 to get 48.

$$48\left(x-1\right)+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Use the distributive property to multiply 48 by x-1.

$$48x-48+12\left(6-3x\right)\left(\frac{1}{2}+\frac{1}{4}\right)^{-2}\geq -4x$$

Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$48x-48+12\left(6-3x\right)\times \left(\frac{3}{4}\right)^{-2}\geq -4x$$

Calculate \frac{3}{4} to the power of -2 and get \frac{16}{9}.

$$48x-48+12\left(6-3x\right)\times \frac{16}{9}\geq -4x$$

Multiply 12 and \frac{16}{9} to get \frac{64}{3}.

$$48x-48+\frac{64}{3}\left(6-3x\right)\geq -4x$$

Use the distributive property to multiply \frac{64}{3} by 6-3x.

$$48x-48+128-64x\geq -4x$$

Add -48 and 128 to get 80.

$$48x+80-64x\geq -4x$$

Combine 48x and -64x to get -16x.

$$-16x+80\geq -4x$$

Add 4x to both sides.

$$-16x+80+4x\geq 0$$

Combine -16x and 4x to get -12x.

$$-12x+80\geq 0$$

Subtract 80 from both sides. Anything subtracted from zero gives its negation.

$$-12x\geq -80$$

Divide both sides by -12. Since -12 is negative, the inequality direction is changed.

$$x\leq \frac{-80}{-12}$$

Reduce the fraction \frac{-80}{-12} to lowest terms by extracting and canceling out -4.

$$x\leq \frac{20}{3}$$