Inequalities
− 6 ≤ x 2 − 5 x ≤ 6
Solve the above inequality.
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1 solution
Do you know the wavy curve method to find solutions of inequalities like these? While it was easy to test a random value of x for the both the double factor inequalities to decide whether the interval is bounded between the two integers of unbounded outside the interval between them, it isn't much efficient to solve for multifactor inequalities. For example, solve for the following two inequalities:
1 . 2 . ( x − 2 ) ( x + 1 ) ( x − 7 ) ( x + 5 ) ( x − 3 ) ≤ 0 x ( x + 5 ) ( x − 7 ) ( x − 2 ) ( x − 4 ) ( x + 3 ) ≥ 0
This is actually two inequalities ,
Look at each one in turn, -6 ≤ x² - 5x and x² - 5x ≤ 6
Inequality 1
− 6 ≤ x 2 − 5 x
⇒ x 2 − 5 x + 6 ≥ 0
x² - 5x + 6 factors to (x - 2)(x - 3)
Therefore, (x - 2)(x - 3) ≥ 0
The equation,(x - 2)(x - 3) = 0
when x = 2 or x = 3
So between 2 and 3, the function will either be
• always greater than zero, or
• always less than zero
To find out which, test a value in between e.g. x = 2.5
2 . 5 2 − 5 × 2 . 5 + 6 = 6 . 2 5 − 1 2 . 5 + 6 = − 0 . 2 5 < 0
Therefore x² - 5x + 6 ≥ 0 for values of x in the intervals ( − ∞ , 2 ] a n d [ 3 , + ∞ )
( − ∞ , 2] ∪ [3, + ∞ )
Inequality 2
x 2 − 5 x ≤ 6
⇒ x 2 − 5 x − 6 ≤ 0
x² - 5x - 6 factors to (x + 1)(x - 6)
Therefore. (x + 1)(x - 6) ≤ 0
The equation. (x + 1)(x - 6) = 0
when x = -1 or x = 6
So between -1 and 6, the function will either be
• always greater than zero, or
• always less than zero
To find out which, test a value in between e.g. x = 0
0 2 − 5 × 0 − 6 = − 6 < 0
So x² - 5x - 6 ≤ 0 for values of x in the interval [-1, 6]
Now, taking the two inequalities together, we have,
{( - ∞ , 2] ∩ [ -1, 6] = [-1,2] ∪ [3, 6]}
Therefore ,
the answer is [ − 1 , 2 ] ∪ [ 3 , 6 ]