Solve x-1/3-2x-1/12-3x-1/2+1geq0

$$\frac { x - 1 } { 3 } - \frac { 2 x - 1 } { 12 } - \frac { 3 x - 1 } { 2 } + 1 \geq 0$$

Solve for x

$$x\leq \frac{15}{16}$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 3,12,2. Since 12 is positive, the inequality direction remains the same.

$$4\left(x-1\right)-\left(2x-1\right)-6\left(3x-1\right)+12\geq 0$$

Use the distributive property to multiply 4 by x-1.

$$4x-4-\left(2x-1\right)-6\left(3x-1\right)+12\geq 0$$

To find the opposite of 2x-1, find the opposite of each term.

$$4x-4-2x-\left(-1\right)-6\left(3x-1\right)+12\geq 0$$

The opposite of -1 is 1.

$$4x-4-2x+1-6\left(3x-1\right)+12\geq 0$$

Combine 4x and -2x to get 2x.

$$2x-4+1-6\left(3x-1\right)+12\geq 0$$

Add -4 and 1 to get -3.

$$2x-3-6\left(3x-1\right)+12\geq 0$$

Use the distributive property to multiply -6 by 3x-1.

$$2x-3-18x+6+12\geq 0$$

Combine 2x and -18x to get -16x.

$$-16x-3+6+12\geq 0$$

Add -3 and 6 to get 3.

$$-16x+3+12\geq 0$$

Add 3 and 12 to get 15.

$$-16x+15\geq 0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$-16x\geq -15$$

Divide both sides by -16. Since -16 is negative, the inequality direction is changed.

$$x\leq \frac{-15}{-16}$$

Fraction \frac{-15}{-16} can be simplified to \frac{15}{16} by removing the negative sign from both the numerator and the denominator.

$$x\leq \frac{15}{16}$$