Question

If for real values of $x$, ${x^2} - 3x + 2 > 0$ and ${x^2} - 3x - 4 \leqslant 0$, then
A. $ - 1 \leqslant x < 1$
B. $ - 1 \leqslant x < 4$
C. $ - 1 \leqslant x < 1\;{\text{or}}\;2 \leqslant x < 4$
D. $2 < x \leqslant 4$

Answer 1

Hint: Rewrite the given equations $x$, ${x^2} - 3x + 2 > 0$ and ${x^2} - 3x - 4 \leqslant 0$ by middle split factoring and then solve accordingly. Use the quadratic formula, get the specific region and range for the given function.

Complete step-by-step answer:
A quadratic equation is an equation that can be arranged in the standard form. This is the combination of variables and constants. The quadratic equation is also defined as the combined form of known and unknowns. The quadratic equation is used to find the curve on a Cartesian grid.
The standard form of quadratic equation is written as,
$a{x^2} + bx + c = 0$
For solving quadratic equations we use quadratic formulas. Quadratic formula is also used to identify the symmetric of Parabola.
A quadratic equation is an equation of second degree. It contains at least one term that is squared. The most important rule of quadratic equations is that the first consonant never be equal to zero.
Quadratic equations are used in our daily life such as, while calculating areas, determining profit, formulating the speed of an object, etc.
The given equation ${x^2} - 2x - x + 2 > 0$ can be written as,
${x^2} - 2x - x + 2 > 0$
$x\left( {x - 2} \right) - 1\left( {x - 2} \right) > 0$
$\left( {x - 1} \right)\left( {x - 2} \right) > 0$
If we break it into region then,
$x \in \left( { - \infty ,1} \right) \cup \left( {2,\infty } \right)$……… (1)
Similarly,
The equation ${x^2} - 3x - 4 \leqslant 0$ can be written as,
${x^2} - 4x + x - 4 \leqslant 0$
$x\left( {x - 4} \right) + 1\left( {x - 4} \right) \leqslant 0$
$\left( {x + 1} \right)\left( {x - 4} \right) \leqslant 0$
If we break it into positive and negative region then,
$x \in \left[ { - 1,4} \right]$ ………(2)
The intersection point is written as,
$\left[ { - 1,4} \right]$
Therefore, the range of $x$ is,
$x \in [ - 1,1) \cup (2,4]$

Hence, the correct option is C.

Note: By middle split factoring the given equation if you face difficulty in factoring the given equation then use a quadratic formula. Make sure that you are not to drop the square root or the plus, minus sign in the middle of your calculations. Numbers in the closed bracket are also included in the solution set. Numbers in the open bracket are not included in the solution set.