Answer

Verified

418.8k+ views

**Hint:**We start solving the problem by finding the interval for all values of x at which the given inequation is defined first. We use the property that for a function $\sqrt{g\left( x \right)}$ is defined only if $g\left( x \right)\ge 0$ for the function $\sqrt{18+x}$. We then use the property $\sqrt{18+x}\ge 0$ to find the interval for the values of x at which the function $2-x$ can be defined. We then find take the inequation $\sqrt{18+x} < 2-x$ and make the necessary calculations and use the property that if the inequality is defined as $\left( x-a \right)\left( x-b \right) > 0$ and $a < b$, then the solution set of the inequality is $x < a\text{ or }x > b$ to find the required solution set.

**Complete step by step answer:**

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 $\sqrt{g\left( x \right)}$ is defined only if $g\left( x \right)\ge 0$.

So, we have $18+x\ge 0$.

$\Rightarrow x\ge -18$.

$\Rightarrow x\in \left[ -18,\infty \right)$ ---(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 $\sqrt{18+x}\ge 0$.

So, we have $2-x > 0$.

$\Rightarrow x < 2$.

$\Rightarrow x\in \left( -\infty ,2 \right)$ ---(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 $x\in \left[ -18,2 \right)$ ---(3).

Now, let us find the solution set for the inequation $\sqrt{18+x} < 2-x$.

\[\Rightarrow {{\left( \sqrt{18+x} \right)}^{2}}<{{\left( 2-x \right)}^{2}}\].

\[\Rightarrow 18+x<4-4x+{{x}^{2}}\].

\[\Rightarrow {{x}^{2}}-5x-14 > 0\].

\[\Rightarrow {{x}^{2}}-7x+2x-14 > 0\].

\[\Rightarrow x\left( x-7 \right)+2\left( x-7 \right) > 0\].

\[\Rightarrow \left( x+2 \right)\left( x-7 \right) > 0\].

\[\Rightarrow \left( x-\left( -2 \right) \right)\left( x-7 \right) > 0\] ---(4).

We know that if the inequality is defined as $\left( x-a \right)\left( x-b \right) > 0$ and $a < b$, then the solution set of the inequality is $x < a\text{ or }x > b$. Using this we get the solution set for inequality in equation (4) as $x < -2\text{ or }x > 7$.

From equation (3), we remove the values of x which were not satisfying the interval $x < -2\text{ or }x > 7$ to get the complete solution set.

So, we get the solution set for the inequality as $x\in \left[ -18,-2 \right)$.

**So, the correct answer is “Option d”.**

**Note:**We should not consider negative square roots while finding the solution set of the given inequalities as it will not always be true which we can see in this problem. We can also solve this problem by drawing the plots of $\sqrt{x+18}$ and $2-x$ as shown below.

From this plot we can see that the values of $\sqrt{x+18}$ lies below the line $2-x$ in the interval $\left[ -18,-2 \right)$.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

10 examples of evaporation in daily life with explanations

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference Between Plant Cell and Animal Cell

What are the monomers and polymers of carbohydrate class 12 chemistry CBSE