Question

Solve inequality in quadratic equations.

| x - 3 | / | x^2 - 4 | <= 1

Find possible real values of x.

Answer 1

I do the answer in detail. Simple way.

given   | x - 3 |  / | x² - 4 |  ≤  1

Clearly  x ≠ 2 or  -2.

Case 1:       | x | < 2,  ie,   -2 < x  < 2

     So  Numerator is negative. Denominator is negative.
=>  (3 - x)/(4 - x²) ≤ 1
=>  4 - x² + x - 3 ≥ 0
=>  x² - x - 1 ≤ 0
=>  Roots = (1+ √5 )/2     valid as it is in the range  |x| < 2
=>  So     (1 - √5)/2  ≤ x ≤ (1+√5)/2

===
case 2 :   2 < x < 3 

      (3-x)/(x² - 4) ≤ 1
      x² - 4 + x - 3 ≥ 0
      x² + x - 7 ≥ 0
      roots:   =  [-1 + √29]/2     ,    (-1-√29) /2 is out of range.
=>    2 < x < [ √29 - 1 ]/2

case 3:   x > 3 

      x - 3 ≤ x² - 4
      x² -x - 1 ≥ 0
     roots :   (1+√5)/2
     So  valid range:   x > (√5 + 1)/2

case 4 : - ∞ < x < -2

      (3-x) ≤ (x²-4)
       x² - 7 + x ≥ 0
    roots:   (-1 + √29)/2  
    so     x < (-1 -√29)/2   or x > (√29 -1)/2

Checking with range of case 4,  we get 
      -∞ < x < (-1 - √29)/2

Answer 2

Step-by-step explanation:

hiii...................