Find the complete solution set for the inequation 18+x < 2-x?

According to the problem, we need to find the complete solution set of the inequation 18+x 0.⇒x 0.⇒x2−7x+2x−14 > 0.⇒x(x−7)+2(x−7) > 0.⇒(x+2)(x−7) > 0.⇒(x−(−2))(x−7) > 0 ---(4).We know that if the inequality is defined as (x−a)(x−b) > 0 and a b. Using this we get the solution set for inequality in equation (4) as x 7.From equation (3), we remove the values of x which were not satisfying the interval x 7 to get the complete solution set.So, we get the solution set for the inequality as x∈[−18,−2).So, the correct answer is “Option 4”.

Find the complete solution set for the inequation 18+x < 2-x?

According to the problem, we need to find the complete solution set of the inequation 18+x 0.⇒x 0.⇒x2−7x+2x−14 > 0.⇒x(x−7)+2(x−7) > 0.⇒(x+2)(x−7) > 0.⇒(x−(−2))(x−7) > 0 ---(4).We know that if the inequality is defined as (x−a)(x−b) > 0 and a b. Using this we get the solution set for inequality in equation (4) as x 7.From equation (3), we remove the values of x which were not satisfying the interval x 7 to get the complete solution set.So, we get the solution set for the inequality as x∈[−18,−2).So, the correct answer is “Option 4”.

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Find the complete solution set for the inequation $\sqrt{18 + x} < 2 - x$ ?

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[−18,−2]

(−∞,−2)∪(7,∞)

(−18,−2)∪(7,∞)

[−18,−2)

answer is D.

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Detailed Solution

According to the problem, we need to find the complete solution set of the inequation

\sqrt{18 + x} < 2 - x .

Let us first find the value for which the given inequation is defined.
We know that for a function

\sqrt{g (x)}

is defined only if g(x)≥0.
So, we have 18+x ≥ 0.
⇒x ≥ −18.
⇒x∈[−18,∞) ---(1).
We know that the 2−x is a polynomial and we can see that all of its values should be greater than zero as we know that the value of the

\sqrt{18 + x}

≥0.
So, we have 2−x > 0.
⇒x < 2.
⇒x∈(−∞,2) ---(2).
The common interval of x for which the given inequation is defined will be the intersection of the values of x obtained in equations (1) and (2).
So, the inequation

\sqrt{18 + x}

< 2−x is defined on x∈[−18,2) ---(3).
Now, let us find the solution set for the inequation

\sqrt{18 + x}

< 2−x.
⇒(

\sqrt{18 + x}

)2 < (2−x)2.
⇒18+x < 4−4x+x2.
⇒x2−5x−14 > 0.
⇒x2−7x+2x−14 > 0.
⇒x(x−7)+2(x−7) > 0.
⇒(x+2)(x−7) > 0.
⇒(x−(−2))(x−7) > 0 ---(4).
We know that if the inequality is defined as (x−a)(x−b) > 0 and a < b, then the solution set of the inequality is x < a or x > b. Using this we get the solution set for inequality in equation (4) as x < −2 or x > 7.
From equation (3), we remove the values of x which were not satisfying the interval x < −2 or x > 7 to get the complete solution set.
So, we get the solution set for the inequality as x∈[−18,−2).
So, the correct answer is “Option 4”.

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