Solve the inequality x - (1 - x)/6 <= (2x + 1)/2 - 3/4.

Step-by-step solution

1. 1

   Combine the left side over the denominator 6. Distributing the minus sign across the numerator is the step most often botched:

   $x-\frac{1-x}{6}=\frac{6x-(1-x)}{6}=\frac{6x-1+x}{6}=\frac{7x-1}{6}$

2. 2

   Combine the right side over the denominator 4.

   $\frac{2x+1}{2}-\frac{3}{4}=\frac{4x+2}{4}-\frac{3}{4}=\frac{4x-1}{4}$

   The inequality is now $\dfrac{7x-1}{6}\le\dfrac{4x-1}{4}$.

3. 3

   Multiply both sides by the LCD 12. Since $12>0$, the direction of $\le$ is unchanged — this is the reason to clear denominators with a positive multiplier rather than, say, $-12$:

   $2(7x-1)\le 3(4x-1)\quad\Longrightarrow\quad 14x-2\le 12x-3$

4. 4

   Isolate $x$. Subtract $12x$, then add $2$, then divide by the positive number $2$:

   $2x-2\le-3\quad\Longrightarrow\quad 2x\le-1\quad\Longrightarrow\quad x\le-\frac{1}{2}$

5. 5

   Test the boundary and one point on each side with exact fractions. At $x=-\tfrac12$ both sides equal $-\tfrac34$, so the endpoint is included. At $x=-1$ the sides are $-\tfrac43$ and $-\tfrac54$, and $-\tfrac43\le-\tfrac54$ holds. At $x=-0.49$ they are $-\tfrac{443}{600}$ and $-\tfrac{37}{50}=-\tfrac{444}{600}$, and $-\tfrac{443}{600}\le-\tfrac{444}{600}$ is false — so the cutoff really is at $-\tfrac12$.

6. 6

   State the solution set.

   $x\in\left(-\infty,\,-\tfrac{1}{2}\right]$