Solve |2x-1/3-3x+1/5-x-2/15|leq5-2x/10

$$| \frac { 2 x - 1 } { 3 } - \frac { 3 x + 1 } { 5 } - \frac { x - 2 } { 15 } | \leq \frac { 5 - 2 x } { 10 }$$

Solve for x

$$x\leq \frac{1}{2}$$

Short Solution Steps

Multiply both sides of the equation by 10. Since 10 is positive, the inequality direction remains the same.

$$10|\frac{2x-1}{3}-\frac{3x+1}{5}-\frac{x-2}{15}|\leq 5-2x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 5 is 15. Multiply \frac{2x-1}{3} times \frac{5}{5}. Multiply \frac{3x+1}{5} times \frac{3}{3}.

$$10|\frac{5\left(2x-1\right)}{15}-\frac{3\left(3x+1\right)}{15}-\frac{x-2}{15}|\leq 5-2x$$

Since \frac{5\left(2x-1\right)}{15} and \frac{3\left(3x+1\right)}{15} have the same denominator, subtract them by subtracting their numerators.

$$10|\frac{5\left(2x-1\right)-3\left(3x+1\right)}{15}-\frac{x-2}{15}|\leq 5-2x$$

Do the multiplications in 5\left(2x-1\right)-3\left(3x+1\right).

$$10|\frac{10x-5-9x-3}{15}-\frac{x-2}{15}|\leq 5-2x$$

Combine like terms in 10x-5-9x-3.

$$10|\frac{x-8}{15}-\frac{x-2}{15}|\leq 5-2x$$

Since \frac{x-8}{15} and \frac{x-2}{15} have the same denominator, subtract them by subtracting their numerators.

$$10|\frac{x-8-\left(x-2\right)}{15}|\leq 5-2x$$

Do the multiplications in x-8-\left(x-2\right).

$$10|\frac{x-8-x+2}{15}|\leq 5-2x$$

Combine like terms in x-8-x+2.

$$10|\frac{-6}{15}|\leq 5-2x$$

Reduce the fraction \frac{-6}{15} to lowest terms by extracting and canceling out 3.

$$10|-\frac{2}{5}|\leq 5-2x$$

The absolute value of a real number a is a when a\geq 0, or -a when a<0. The absolute value of -\frac{2}{5} is \frac{2}{5}.

$$10\times \frac{2}{5}\leq 5-2x$$

Express 10\times \frac{2}{5} as a single fraction.

$$\frac{10\times 2}{5}\leq 5-2x$$

Multiply 10 and 2 to get 20.

$$\frac{20}{5}\leq 5-2x$$

Divide 20 by 5 to get 4.

$$4\leq 5-2x$$

Swap sides so that all variable terms are on the left hand side. This changes the sign direction.

$$5-2x\geq 4$$

Subtract 5 from both sides.

$$-2x\geq 4-5$$

Subtract 5 from 4 to get -1.

$$-2x\geq -1$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\leq \frac{-1}{-2}$$

Fraction \frac{-1}{-2} can be simplified to \frac{1}{2} by removing the negative sign from both the numerator and the denominator.

$$x\leq \frac{1}{2}$$