Solve 6(8+1/8-2x-3/16)leq3(3/4x-1/4)-frac{3}{8}(3x-2)

$$6 ( \frac { 8 + 1 } { 8 } - \frac { 2 x - 3 } { 16 } ) \leq 3 ( \frac { 3 } { 4 } x - \frac { 1 } { 4 } ) - \frac { 3 } { 8 } ( 3 x - 2 )$$

Solve for x

$$x\geq \frac{21}{5}$$

Short Solution Steps

Multiply both sides of the equation by 16, the least common multiple of 8,16,4. Since 16 is positive, the inequality direction remains the same.

$$96\left(\frac{8+1}{8}-\frac{2x-3}{16}\right)\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Add 8 and 1 to get 9.

$$96\left(\frac{9}{8}-\frac{2x-3}{16}\right)\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8 and 16 is 16. Multiply \frac{9}{8} times \frac{2}{2}.

$$96\left(\frac{9\times 2}{16}-\frac{2x-3}{16}\right)\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Since \frac{9\times 2}{16} and \frac{2x-3}{16} have the same denominator, subtract them by subtracting their numerators.

$$96\times \frac{9\times 2-\left(2x-3\right)}{16}\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Do the multiplications in 9\times 2-\left(2x-3\right).

$$96\times \frac{18-2x+3}{16}\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Combine like terms in 18-2x+3.

$$96\times \frac{21-2x}{16}\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Cancel out 16, the greatest common factor in 96 and 16.

$$6\left(21-2x\right)\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Use the distributive property to multiply 6 by 21-2x.

$$126-12x\leq 48\left(\frac{3}{4}x-\frac{1}{4}\right)-6\left(3x-2\right)$$

Use the distributive property to multiply 48 by \frac{3}{4}x-\frac{1}{4}.

$$126-12x\leq 48\times \frac{3}{4}x+48\left(-\frac{1}{4}\right)-6\left(3x-2\right)$$

Express 48\times \frac{3}{4} as a single fraction.

$$126-12x\leq \frac{48\times 3}{4}x+48\left(-\frac{1}{4}\right)-6\left(3x-2\right)$$

Multiply 48 and 3 to get 144.

$$126-12x\leq \frac{144}{4}x+48\left(-\frac{1}{4}\right)-6\left(3x-2\right)$$

Divide 144 by 4 to get 36.

$$126-12x\leq 36x+48\left(-\frac{1}{4}\right)-6\left(3x-2\right)$$

Express 48\left(-\frac{1}{4}\right) as a single fraction.

$$126-12x\leq 36x+\frac{48\left(-1\right)}{4}-6\left(3x-2\right)$$

Multiply 48 and -1 to get -48.

$$126-12x\leq 36x+\frac{-48}{4}-6\left(3x-2\right)$$

Divide -48 by 4 to get -12.

$$126-12x\leq 36x-12-6\left(3x-2\right)$$

Use the distributive property to multiply -6 by 3x-2.

$$126-12x\leq 36x-12-18x+12$$

Combine 36x and -18x to get 18x.

$$126-12x\leq 18x-12+12$$

Add -12 and 12 to get 0.

$$126-12x\leq 18x$$

Subtract 18x from both sides.

$$126-12x-18x\leq 0$$

Combine -12x and -18x to get -30x.

$$126-30x\leq 0$$

Subtract 126 from both sides. Anything subtracted from zero gives its negation.

$$-30x\leq -126$$

Divide both sides by -30. Since -30 is negative, the inequality direction is changed.

$$x\geq \frac{-126}{-30}$$

Reduce the fraction \frac{-126}{-30} to lowest terms by extracting and canceling out -6.

$$x\geq \frac{21}{5}$$