Solve the inequality x - 2x + 2 >= 1.

Step-by-step solution

1. 1

   Combine the two $x$ terms. They have coefficients $+1$ and $-2$, so their sum is negative:

   $x-2x=-x\quad\Longrightarrow\quad -x+2\ge 1$

   Recognising the negative coefficient early tells you a sign flip is coming.

2. 2

   Isolate the variable term. Subtract $2$ from both sides:

   $-x\ge -1$

3. 3

   Divide (or multiply) by $-1$ and reverse the direction.

   $x\le 1$

   Multiplying by $-1$ reflects the number line, which reverses the order of every pair of numbers — hence $\ge$ becomes $\le$.

4. 4

   Verify with test points. At $x=0$: $0-0+2=2\ge 1$ $\checkmark$. At $x=5$: $5-10+2=-3\ge 1$ is false $\checkmark$. At the boundary $x=1$: $1-2+2=1\ge 1$ $\checkmark$, so the endpoint is included and the solution set is $(-\infty,1]$.