[tex]\sf \displaystyle log_{0.5} \ log_{5} (x^{2} - 4) > log_{0.5} (1)[/tex]
We can write expression as,
[tex]\sf \displaystyle log_{5} (x^{2} - 4) < 1.[/tex]
[tex]\sf \displaystyle x^{2} - 4 < (5)^{1}.[/tex]
[tex]\sf \displaystyle x^{2} - 4 < 5.[/tex]
[tex]\sf \displaystyle x^{2} - 4 - 5 < 0.[/tex]
[tex]\sf \displaystyle x^{2} -9 < 0.[/tex]
[tex]\sf \displaystyle (x + 3)(x - 3) < 0.[/tex]
Zeroes are : x = 3 and x = - 3.
Put the point on wavy curve method, we get.
⇒ x ∈ (-3, 3). - - - - - (1).
For log defined : ㏒₅(x² - 4) > 0.
⇒ ㏒₅(x² - 4) > 0.
⇒ x² - 4 > 5⁰.
⇒ x² - 4 > 1.
⇒ x² - 4 - 1 > 0.
⇒ x² - 5 > 0.
⇒ (x + √5)(x - √5) > 0.
Zeroes of the expression is,
⇒ x = - √5 and x = √5.
⇒ x ∈ (- ∞, - √5) ∪ (√5, ∞). - - - - - (2).
Taking intersection of expression (1) and expression (2), we get.
⇒ x ∈ (- 3, - √5) ∪ (√5, 3).
∴ Option [A] is correct answer.
Rules of inequality.
We can multiply Or divide any non - zero number "k" on both sides of inequality sign of inequality will change according to sign of k that is,
If k > 0 then sign of inequality will remain same.
If k < 0 then sign of inequality will get reversed.
Wavy curve method.
Step - 1 : Check structure.
Step - 2 : Find critical points.
Step - 3 : Plot all critical points on number lines.
Step - 4 : Check sign of given example by putting value of x from different parts of number line.
[tex]\huge\purple {\mathfrak{Answer }}[/tex] So,
I hope it helps you!!
have a Bangtan Day!!^^
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Verified answer
EXPLANATION.
[tex]\sf \displaystyle log_{0.5} \ log_{5} (x^{2} - 4) > log_{0.5} (1)[/tex]
We can write expression as,
[tex]\sf \displaystyle log_{5} (x^{2} - 4) < 1.[/tex]
[tex]\sf \displaystyle x^{2} - 4 < (5)^{1}.[/tex]
[tex]\sf \displaystyle x^{2} - 4 < 5.[/tex]
[tex]\sf \displaystyle x^{2} - 4 - 5 < 0.[/tex]
[tex]\sf \displaystyle x^{2} -9 < 0.[/tex]
[tex]\sf \displaystyle (x + 3)(x - 3) < 0.[/tex]
Zeroes are : x = 3 and x = - 3.
Put the point on wavy curve method, we get.
⇒ x ∈ (-3, 3). - - - - - (1).
For log defined : ㏒₅(x² - 4) > 0.
⇒ ㏒₅(x² - 4) > 0.
⇒ x² - 4 > 5⁰.
⇒ x² - 4 > 1.
⇒ x² - 4 - 1 > 0.
⇒ x² - 5 > 0.
⇒ (x + √5)(x - √5) > 0.
Zeroes of the expression is,
⇒ x = - √5 and x = √5.
Put the point on wavy curve method, we get.
⇒ x ∈ (- ∞, - √5) ∪ (√5, ∞). - - - - - (2).
Taking intersection of expression (1) and expression (2), we get.
⇒ x ∈ (- 3, - √5) ∪ (√5, 3).
∴ Option [A] is correct answer.
MORE INFORMATION.
Rules of inequality.
We can multiply Or divide any non - zero number "k" on both sides of inequality sign of inequality will change according to sign of k that is,
If k > 0 then sign of inequality will remain same.
If k < 0 then sign of inequality will get reversed.
Wavy curve method.
Step - 1 : Check structure.
Step - 2 : Find critical points.
Step - 3 : Plot all critical points on number lines.
Step - 4 : Check sign of given example by putting value of x from different parts of number line.
[tex]\huge\purple {\mathfrak{Answer }}[/tex] So,
Zeroes are : x = 3 and x = - 3.
Put the point on wavy curve method, we get.
⇒ x ∈ (-3, 3). - - - - - (1).
For log defined : ㏒₅(x² - 4) > 0.
⇒ ㏒₅(x² - 4) > 0.
⇒ x² - 4 > 5⁰.
⇒ x² - 4 > 1.
⇒ x² - 4 - 1 > 0.
⇒ x² - 5 > 0.
⇒ (x + √5)(x - √5) > 0.
Zeroes of the expression is,
⇒ x = - √5 and x = √5.
Put the point on wavy curve method, we get.
⇒ x ∈ (- ∞, - √5) ∪ (√5, ∞). - - - - - (2).
Taking intersection of expression (1) and expression (2), we get.
⇒ x ∈ (- 3, - √5) ∪ (√5, 3).
∴ Option [A] is correct answer.
I hope it helps you!!
have a Bangtan Day!!^^