Solve the inequality (4/3)(-x + 2) - (x + 4)/6 > x/2 + 2.

Step-by-step solution

1. 1

   Choose a common denominator before expanding. The denominators present are $3$, $6$ and $2$, whose least common multiple is $6$. Rewriting everything over $6$ first keeps you from distributing fractions term by term, which is where sign slips usually happen:

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

2. 2

   Combine the two fractions on the left. They already share the denominator $6$:

   $\frac{-8x+16}{6}-\frac{x+4}{6}=\frac{-8x+16-x-4}{6}=\frac{-9x+12}{6}$

   Note the minus sign distributes over both $x$ and $4$.

3. 3

   Multiply both sides by 6. Since $6>0$, multiplying does not change the direction of the inequality:

   $-9x+12>6\left(\frac{x}{2}+2\right)=3x+12$

4. 4

   Collect the variable terms. Subtract $3x$ and subtract $12$ from both sides:

   $-9x-3x>12-12\quad\Longrightarrow\quad -12x>0$

5. 5

   Divide by a negative number and flip the sign. Dividing both sides by $-12$ reverses the inequality:

   $x<0$

   $\boxed{x<0\quad\text{i.e. }x\in(-\infty,0)}$

6. 6

   Test one value on each side. At $x=-1$ the left side is $\tfrac{4}{3}(3)-\tfrac{3}{6}=4-0.5=3.5$ and the right side is $-0.5+2=1.5$, so $3.5>1.5$ holds. At $x=1$ the left side is $\tfrac{4}{3}(1)-\tfrac{5}{6}=\tfrac{8}{6}-\tfrac{5}{6}=0.5$ and the right side is $2.5$, so the inequality fails. Both tests agree with $x<0$.