Solve the inequality x/3 > 4 - (x - 2)/2 and give the solution in interval notation.

Step-by-step solution

1. 1

   Simplify the right-hand side before moving anything. The subtraction applies to the whole fraction, so split it first:

   $\frac{x-2}{2}=\frac{x}{2}-1,$

   and therefore

   $4-\frac{x-2}{2}=4-\left(\frac{x}{2}-1\right)=4-\frac{x}{2}+1=5-\frac{x}{2}.$

   The classic error is writing $4-\frac{x}{2}-1=3-\frac{x}{2}$: the minus in front must reach the $-1$ as well and turn it into $+1$.

2. 2

   Rewrite the inequality with the simplified right side. The problem is now

   $\frac{x}{3}>5-\frac{x}{2}.$

   Both $x$ terms have simple denominators, so it is cheapest to collect them rather than multiply everything by $6$ straight away — either route works, but collecting keeps the numbers small.

3. 3

   Collect the $x$ terms on the left. Add $\frac{x}{2}$ to both sides and use the common denominator $6$:

   $\frac{x}{3}+\frac{x}{2}=\frac{2x}{6}+\frac{3x}{6}=\frac{5x}{6},$

   so the inequality becomes

   $\frac{5x}{6}>5.$

   Adding a term to both sides never changes the direction of an inequality, so the $>$ is untouched.

4. 4

   Divide by the positive coefficient. Multiplying both sides by $\tfrac{6}{5}$ (a positive number, so the sign stays):

   $x>5\cdot\frac{6}{5}=6.$

   In interval notation the solution is $(6,\infty)$, with $6$ excluded because the inequality is strict.

5. 5

   Test either side of the boundary. At $x=6.1$: left $=2.0333$, right $=4-\frac{4.1}{2}=1.95$, and $2.0333>1.95$ ✓. At $x=5.9$: left $=1.9667$, right $=4-1.95=2.05$, so the inequality fails ✓. At $x=6$ exactly both sides equal $2$, confirming $6$ is the boundary and correctly excluded.