Solve the inequality -(1/3)(x - 7) >= (1/2)(3x + 1).

Step-by-step solution

1. 1

   Clear both denominators in one move. The denominators are $3$ and $2$, so multiply every term by their LCM $6$. Because $6>0$, the inequality direction is unaffected — this is the safe way to remove fractions:

   $6\cdot\left[-\frac13(x-7)\right]\ge6\cdot\left[\frac12(3x+1)\right]$

   $-2(x-7)\ge3(3x+1)$

2. 2

   Distribute on both sides.

   $-2x+14\ge9x+3$

   The $-2$ multiplies both terms inside the bracket, turning $-7$ into $+14$.

3. 3

   Collect the variable on one side. Subtract $9x$ from both sides (adding or subtracting never flips the sign):

   $-11x+14\ge3$

   then subtract $14$:

   $-11x\ge-11$

4. 4

   Divide by the negative coefficient and flip. Dividing both sides by $-11$ reverses the inequality:

   $x\le1$

   This single sign reversal is the entire trap in the problem; keeping $\ge$ here would give the exactly wrong half-line.

5. 5

   Test one point on each side of the boundary. At $x=1$: left $=-\tfrac13(-6)=2$, right $=\tfrac12(4)=2$, so equality holds ✓ (the endpoint is included). At $x=0$: left $=\tfrac73\approx2.33$, right $=0.5$, and $2.33\ge0.5$ is true ✓. At $x=2$: left $=-\tfrac53\approx-1.67$, right $=3.5$, and $-1.67\ge3.5$ is false ✓. The solution set is $(-\infty,1]$.