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How do you solve \[\dfrac{1}{4}n+12\ge \dfrac{3}{4}n-4\] and graph the solution on a number line?

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Last updated date: 24th Jul 2024
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Answer
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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.
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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.