Solve (x-1/2-x-1/4)-x+8/3leq1/2(x-9)

$$( \frac { x - 1 } { 2 } - \frac { x - 1 } { 4 } ) - \frac { x + 8 } { 3 } \leq \frac { 1 } { 2 } ( x - 9 )$$

Solve for x

$$x\geq \frac{19}{7}$$

Solution Steps

Multiply both sides of the equation by 12, the least common multiple of 2,4,3. Since 12 is positive, the inequality direction remains the same.

$$6\left(x-1\right)-3\left(x-1\right)-4\left(x+8\right)\leq 6\left(x-9\right)$$

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

$$6x-6-3\left(x-1\right)-4\left(x+8\right)\leq 6\left(x-9\right)$$

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

$$6x-6-3x+3-4\left(x+8\right)\leq 6\left(x-9\right)$$

Combine 6x and -3x to get 3x.

$$3x-6+3-4\left(x+8\right)\leq 6\left(x-9\right)$$

Add -6 and 3 to get -3.

$$3x-3-4\left(x+8\right)\leq 6\left(x-9\right)$$

Use the distributive property to multiply -4 by x+8.

$$3x-3-4x-32\leq 6\left(x-9\right)$$

Combine 3x and -4x to get -x.

$$-x-3-32\leq 6\left(x-9\right)$$

Subtract 32 from -3 to get -35.

$$-x-35\leq 6\left(x-9\right)$$

Use the distributive property to multiply 6 by x-9.

$$-x-35\leq 6x-54$$

Subtract 6x from both sides.

$$-x-35-6x\leq -54$$

Combine -x and -6x to get -7x.

$$-7x-35\leq -54$$

Add 35 to both sides.

$$-7x\leq -54+35$$

Add -54 and 35 to get -19.

$$-7x\leq -19$$

Divide both sides by -7. Since -7 is negative, the inequality direction is changed.

$$x\geq \frac{-19}{-7}$$

Fraction \frac{-19}{-7} can be simplified to \frac{19}{7} by removing the negative sign from both the numerator and the denominator.

$$x\geq \frac{19}{7}$$