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 $1 < x < 2$.
Important Questions from Mathematical Inequalities
A stick of length one meter is broken at two locations at distances of $b_1$ and $b_2$ from the origin (0), as shown in the figure. Note that $0 < b_1 < b_2< 1$. Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?
Note: All lengths are in meter. The figure shown is representative.
Consider the following inequalities.
(i) $3p - q < 4$
(ii) $3q - p < 12$
Which one of the following expressions below satisfies the above two inequalities?Consider the following inequalities.
(i) $2x - 1 > 7$
(ii) $2x - 9 < 1$
Which one of the following expressions below satisfies the above two inequalities?Which one of the following is a representation (not to scale and in bold) of all values of $x$ satisfying the inequality $2 – 5x \le \frac{6x-5}{3}$on the real numberline?
- The number of solutions for the following system of inequalities is
$X_1 \ge 0$
$X_2 \ge 0$
$X_1+ X_2 \le 10$
$2X_1+ 2X_2 \ge 22$
