Solving the Inequality $x^2 -3x+2 < 0$

To find the range of values for $x$ satisfying $x^2 -3x+2 < 0$, we first find the roots of the equation $x^2 -3x+2 = 0$.

Finding the Roots

Factor the quadratic expression:

$ x^2 -3x+2 = (x-1)(x-2) $

Set the factored expression to zero:

$ (x-1)(x-2) = 0 $

The roots are $x=1$ and $x=2$. These roots divide the number line into three intervals: $x < 1$, $1 < x < 2$, and $x > 2$.

Testing Intervals

We test the sign of $(x-1)(x-2)$ in each interval to find where it is less than zero ($< 0$):

• For $x < 1$: Choose $x=0$. $(0-1)(0-2) = (-1)(-2) = 2$. Since $2 \not< 0$, this interval is not a solution.
• For $1 < x < 2$: Choose $x=1.5$. $(1.5-1)(1.5-2) = (0.5)(-0.5) = -0.25$. Since $-0.25 < 0$, this interval is the solution.
• For $x > 2$: Choose $x=3$. $(3-1)(3-2) = (2)(1) = 2$. Since $2 \not< 0$, this interval is not a solution.

Conclusion

The inequality $x^2 -3x+2 < 0$ is satisfied when $1 < x < 2$.