Solve 1/3(1/2-3/4x)+21/2(frac{1}{5}x-1frac{1}{5})geq2frac{1}{6}x-1frac{1}{3}(frac{1}{2}x-1)

$$\frac { 1 } { 3 } ( \frac { 1 } { 2 } - \frac { 3 } { 4 } x ) + 2 \frac { 1 } { 2 } ( \frac { 1 } { 5 } x - 1 \frac { 1 } { 5 } ) \geq 2 \frac { 1 } { 6 } x - 1 \frac { 1 } { 3 } ( \frac { 1 } { 2 } x - 1 )$$

Solve for x

$$x\leq -\frac{10}{3}$$

Solution Steps

Multiply both sides of the equation by 60, the least common multiple of 3,2,4,5,6. Since 60 is positive, the inequality direction remains the same.

$$20\left(\frac{1}{2}-\frac{3}{4}x\right)+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Use the distributive property to multiply 20 by \frac{1}{2}-\frac{3}{4}x.

$$20\times \frac{1}{2}+20\left(-\frac{3}{4}\right)x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 20 and \frac{1}{2} to get \frac{20}{2}.

$$\frac{20}{2}+20\left(-\frac{3}{4}\right)x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Divide 20 by 2 to get 10.

$$10+20\left(-\frac{3}{4}\right)x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Express 20\left(-\frac{3}{4}\right) as a single fraction.

$$10+\frac{20\left(-3\right)}{4}x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 20 and -3 to get -60.

$$10+\frac{-60}{4}x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Divide -60 by 4 to get -15.

$$10-15x+30\left(2\times 2+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 2 and 2 to get 4.

$$10-15x+30\left(4+1\right)\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Add 4 and 1 to get 5.

$$10-15x+30\times 5\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 30 and 5 to get 150.

$$10-15x+150\left(\frac{1}{5}x-\frac{1\times 5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 1 and 5 to get 5.

$$10-15x+150\left(\frac{1}{5}x-\frac{5+1}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Add 5 and 1 to get 6.

$$10-15x+150\left(\frac{1}{5}x-\frac{6}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Use the distributive property to multiply 150 by \frac{1}{5}x-\frac{6}{5}.

$$10-15x+150\times \frac{1}{5}x+150\left(-\frac{6}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 150 and \frac{1}{5} to get \frac{150}{5}.

$$10-15x+\frac{150}{5}x+150\left(-\frac{6}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Divide 150 by 5 to get 30.

$$10-15x+30x+150\left(-\frac{6}{5}\right)\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Express 150\left(-\frac{6}{5}\right) as a single fraction.

$$10-15x+30x+\frac{150\left(-6\right)}{5}\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 150 and -6 to get -900.

$$10-15x+30x+\frac{-900}{5}\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Divide -900 by 5 to get -180.

$$10-15x+30x-180\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Combine -15x and 30x to get 15x.

$$10+15x-180\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Subtract 180 from 10 to get -170.

$$-170+15x\geq 10\left(2\times 6+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 2 and 6 to get 12.

$$-170+15x\geq 10\left(12+1\right)x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Add 12 and 1 to get 13.

$$-170+15x\geq 10\times 13x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 10 and 13 to get 130.

$$-170+15x\geq 130x-20\left(1\times 3+1\right)\left(\frac{1}{2}x-1\right)$$

Multiply 1 and 3 to get 3.

$$-170+15x\geq 130x-20\left(3+1\right)\left(\frac{1}{2}x-1\right)$$

Add 3 and 1 to get 4.

$$-170+15x\geq 130x-20\times 4\left(\frac{1}{2}x-1\right)$$

Multiply 20 and 4 to get 80.

$$-170+15x\geq 130x-80\left(\frac{1}{2}x-1\right)$$

Use the distributive property to multiply -80 by \frac{1}{2}x-1.

$$-170+15x\geq 130x-80\times \frac{1}{2}x+80$$

Multiply -80 and \frac{1}{2} to get \frac{-80}{2}.

$$-170+15x\geq 130x+\frac{-80}{2}x+80$$

Divide -80 by 2 to get -40.

$$-170+15x\geq 130x-40x+80$$

Combine 130x and -40x to get 90x.

$$-170+15x\geq 90x+80$$

Subtract 90x from both sides.

$$-170+15x-90x\geq 80$$

Combine 15x and -90x to get -75x.

$$-170-75x\geq 80$$

Add 170 to both sides.

$$-75x\geq 80+170$$

Add 80 and 170 to get 250.

$$-75x\geq 250$$

Divide both sides by -75. Since -75 is negative, the inequality direction is changed.

$$x\leq \frac{250}{-75}$$

Reduce the fraction \frac{250}{-75} to lowest terms by extracting and canceling out 25.

$$x\leq -\frac{10}{3}$$