Solve the inequality 5 minus one half of (x + 4) is greater than or equal to 4.

Step-by-step solution

1. 1

   Distribute the $-\frac12$ over the bracket. Both terms inside are affected:

   $5-\frac12x-2\ge 4$

   since $-\frac12\cdot 4=-2$. Keeping the minus sign attached to the fraction is what makes the next steps come out right.

2. 2

   Combine the constants on the left.

   $5-2=3\quad\Rightarrow\quad 3-\frac12x\ge 4$

3. 3

   Isolate the $x$ term. Subtract $3$ from both sides — an addition, so the direction is untouched:

   $-\frac12x\ge 1$

4. 4

   Multiply by $-2$ and reverse the inequality. This is the crux: multiplying or dividing by a negative number flips $\ge$ into $\le$:

   $x\le -2$

   Leaving it as $x\ge -2$ would give exactly the wrong half of the line.

5. 5

   Test three points to be sure. At $x=-2$: $5-\frac12(2)=4$, and $4\ge 4$ holds (so the endpoint is included). At $x=-4$: $5-0=5\ge 4$ ✓. At $x=0$: $5-2=3$, and $3\ge 4$ is false ✓. So the solution is $x\le -2$, i.e. $x\in(-\infty,-2]$.