Solve the inequality 4(x + 1) is greater than or equal to x + 2(x - 1).

Step-by-step solution

1. 1

   Expand each side independently. Distribute before comparing anything:

   $4(x+1)=4x+4,\qquad x+2(x-1)=x+2x-2=3x-2.$

   On the right, the $2$ multiplies only the bracket; the leading $x$ is a separate term.

2. 2

   Rewrite the inequality with both sides simplified.

   $4x+4\ge 3x-2.$

3. 3

   Collect the x-terms on one side. Subtract $3x$ from both sides:

   $x+4\ge -2.$

   Adding or subtracting the same quantity never changes the direction of an inequality.

4. 4

   Isolate x. Subtract $4$:

   $x\ge -6.$

5. 5

   Note why the sign did not flip. The direction reverses only when you multiply or divide by a negative number. Here every step was an addition or subtraction, so $\ge$ stayed $\ge$ throughout.

6. 6

   Test one value inside and one outside. At $x=0$: $4(1)=4$ and $0+2(-1)=-2$, and $4\ge -2$ ✓. At $x=-10$: $4(-9)=-36$ and $-10+2(-11)=-32$, and $-36\ge -32$ is false ✓. The boundary $x=-6$ gives $-20\ge -20$, true, so it is included.