Question

Plz answer thr ques correctly⇪​

Answer 1

x²-10<=6,x²<=6+10,x²<=√16,x=+4,-4

Answer 2

Answér :

(iv). (-∞,-4] U [-2,2] U [4,∞)

Note :

• If |x| = a , then x = ± a

• If |x| < a , then x € (-a,a) [ OR -a < x < a ]

• If |x| ≤ a , then x € [-a,a] [ OR -a ≤ x ≤ a ]

• If |x| > a , then x € (-∞,-a) U (a,∞)

[ OR x < -a or x > a ]

• If |x| ≥ a , then x € (-∞,-a] U [a,∞)

[ OR x ≤ -a or x ≥ a ]

Solution :

• Given : |x² - 10| ≤ 6 , x € R
• To find : Solution set

We have ;

|x² - 10| ≤ 6

Here ,

Two cases arises ;

1). x² - 10 ≤ -6

OR

2). x² - 10 ≥ 6

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Case1 : x² - 10 ≤ -6

=> x² - 10 ≤ -6

=> x² - 10 + 6 ≤ 0

=> x² - 4 ≤ 0

=> (x - 2)(x + 2) ≤ 0

Here ,

Two sub-cases arises ;

1a). x - 2 ≥ 0 and x + 2 ≤ 0

OR

1b). x - 2 ≤ 0 and x + 2 ≥ 0

Case1a : x - 2 ≥ 0 and x + 2 ≤ 0

=> x - 2 ≥ 0 and x + 2 ≤ 0

=> x ≥ 2 and x ≤ -2

=> x € [2,∞) and x € (-∞,-2]

=> x € [2,∞) ∩ (-∞,-2]

=> x € ∅ [ °•° There exist no real number which is greater than 2 and smaller than -2 simultaneously ]

OR

Case1b : x - 2 ≤ 0 and x + 2 ≥ 0

=> x - 2 ≤ 0 and x + 2 ≥ 0

=> x ≤ 2 and x ≥ -2

=> x € (-∞,2] and x € [-2,∞)

=> x € (-∞,2] ∩ [-2,∞)

=> x € [-2,2] [ or -2 ≤ x ≤ 2 ]

Here ,

The solution set for case1 will be given as the union of solutions found in both the sub-cases .

Thus ,

=> x € ∅ or x € [-2,2]

=> x € ∅ U [-2,2]

=> x € [-2,2]

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OR

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Case2 : x² - 10 ≥ 6

=> x² - 10 ≥ 6

=> x² - 10 - 6 ≥ 0

=> x² - 16 ≥ 0

=> (x - 4)(x + 4) ≥ 0

Here ,

Two sub-cases arises ;

2a). x - 4 ≥ 0 and x + 4 ≥ 0

OR

2b). x - 4 ≤ 0 and x + 4 ≤ 0

Case2a : x - 4 ≥ 0 and x + 4 ≥ 0

=> x - 4 ≥ 0 and x + 4 ≥ 0

=> x ≥ 4 and x ≥ -4

=> x € [4,∞) and x € [-4,∞)

=> x € [4,∞) ∩ [-4,∞)

=> x € [4,∞)

OR

Case2b : x - 4 ≤ 0 and x + 4 ≤ 0

=> x - 4 ≤ 0 and x + 4 ≤ 0

=> x ≤ 4 and x ≤ -4

=> x € (-∞,4] and x € (-∞,-4]

=> x € (-∞,4] ∩ (-∞,-4]

=> x € (-∞,-4]

Here ,

The solution set for case2 will be given as the union of solutions found in both the sub-cases .

Thus ,

=> x € [4,∞) or x € (-∞,-4]

=> x € [4,∞) U (-∞,-4]

=> x € (-∞,-4] U [4,∞)

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Here ,

The solution set of the given inequation will be given as the union of the solutions found in both the cases .

Thus ,

Solution set will be given as ;

=> x € [-2,2] U (-∞,-4] U [4,∞)

=> x € (-∞,-4] U [-2,2] U [4,∞)

Hence ,

The solution set is :

(-∞,-4] U [-2,2] U [4,∞)