Answer

Verified

384.9k+ views

**Hint:**Multiply both the sides of the given inequality with 4 to remove the fractional terms. Do not change the inequality sign. Now, simplify the inequation by rearranging the terms, take the terms containing the variable ‘n’ to the left-hand side and the constant terms to the right-hand side. Simplify both the sides with simple addition and subtraction. Make the coefficient of n equal to 1 and change the direction of inequality if you divide or multiply both the sides with a negative number. Show the set of values of ‘n’ on a number line extending from \[-\infty \] to \[\infty \].

**Complete step-by-step solution:**

Here, we have been provided with the inequality \[\dfrac{1}{4}n+12\ge \dfrac{3}{4}n-4\] and we are asked to solve it and represent it on a number line. So, let us solve the values of n.

\[\because \dfrac{1}{4}n+12\ge \dfrac{3}{4}n-4\]

Multiplying both the sides with 4, we get,

\[\Rightarrow n+48\ge 3n-16\]

Here, as you can see that the direction of inequality does not change, this is because we have multiplied both the sides with a positive number.

Now, rearranging the terms by taking the terms containing the variable ‘n’ to the L.H.S. and the constant terms to the R.H.S., we get,

\[\begin{align}

& \Rightarrow n-3n\ge -48-16 \\

& \Rightarrow -2n\ge -64 \\

\end{align}\]

Dividing both the sides with (-2), we get,

\[\Rightarrow n\le 32\]

As you can see here the direction of inequality gets reversed, this is because we have divided both the sides with a negative number.

Hence, our solution is \[n\le 32\]. This can be represented in interval or set form as \[n\in \left( -\infty ,32 \right]\].

Now, let us represent the solution set of ‘n’ on a number line. Since, a number line extends from \[-\infty \] to \[\infty \], so marking the value 32 on it we have, n must be less than or equal to 32.

Hence, the above number line represents our answer.

**Note:**One may note that we have a closed interval on 32 and an open interval on \[-\infty \], this is because the value of ‘n’ can also be 32. The value of n is tending to \[-\infty \], since \[\infty \] and \[-\infty \] are not real numbers so we use an open interval while writing them in set form. Now, if we are asked to draw the graph of \[n\le 32\] then we will consider n equal to x on a cartesian plane and shade the left part of the line x = 32 as our answer. You must remember the rules of changing the direction of inequality sign otherwise you may get the wrong answer.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Guru Purnima speech in English in 100 words class 7 english CBSE

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

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Difference Between Plant Cell and Animal Cell

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

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers