Solve the inequality -x^2 + 4x - 3 >= 0.

Step-by-step solution

1. 1

   Make the leading coefficient positive. Multiply the whole inequality by $-1$ and reverse the direction:

   $x^2-4x+3\le 0$

   Skipping the flip is the single most common error with a negative $x^2$ coefficient; it would give the complement of the true answer.

2. 2

   Factor. Product $3$, sum $-4$ gives $-1$ and $-3$:

   $(x-1)(x-3)\le 0$

3. 3

   Read off the region. The parabola $y=x^2-4x+3$ opens upward, so it is at or below the axis exactly between its roots, endpoints included:

   $1\le x\le 3$

4. 4

   Sanity-check against the original inequality. $y=-x^2+4x-3$ opens downward with the same roots $1$ and $3$, so it is at or above the axis between them — the same interval, reached without any flipping. Two independent routes agreeing is a good check.

5. 5

   Test values. At $x=2$: $-4+8-3=1\ge 0$ $\checkmark$. At $x=0$: $-3\not\ge 0$ $\checkmark$. At $x=4$: $-16+16-3=-3\not\ge 0$ $\checkmark$. At $x=1$ and $x=3$ the value is $0$, which satisfies $\ge 0$ $\checkmark$.