Question

Find the number of integral solutions of: $$|x^2-1|+|x^2-5x+6|=|5x-7|$$.

Answer 1

Hint: Some basic formulae to solve such questions:
1.The property of modulus $$|a|+|b|⩽|a-b|$$
2.Use splitting the middle term method to solve quadratic equations.

Complete step-by-step answer:
We can clearly see that above equation is in the form of $$|a|+|b|⩽|a-b|$$.
Where,
 $$|a|=|x^2-1|$$;
$$|b|=|x^2-5x+6|$$;
$$|a-b|=|x^2-1-(x^2-5x+6)|\Rightarrow |a-b|=|5x-7|$$
$$\begin{gathered} \Rightarrow (x^2-1)+(x^2-5x+6)⩽(5x-7) \\ \Rightarrow 2x^2-5x+5⩽5x-7 \\ \Rightarrow 2x^2-10x+12⩽0 \\ \Rightarrow x^2-5x+6⩽0 \\ \Rightarrow x^2-3x-2x+6⩽0 \\ \Rightarrow x(x-3)-2(x-3)⩽0 \\ \Rightarrow (x-3)(x-2)⩽0 \\ \Rightarrow 2⩽x⩽3 \end{gathered}$$
Required integrals solutions: $$2⩽x⩽3$$.

Note: Two bracket linear equalities must be known to student’s to solve such kinds of questions.
A quadratic equation can at most give two solutions. The solutions can be real and distinct; real and equal or in complex form depending upon the value of D is greater than 0, equals 0 or less than 0.