Question

How to solve the inequality \[-23\le 2x-1\le 1\]?

Answer 1

Hint: To solve this type of question that has been mentioned above first we are going to divide the whole inequality into two inequality first being the inequality with more than and the other being with the less than inequality.

Complete step-by-step solution:
For the type of question that has been mentioned above as there are two inequality signs i.e. greater than equal to and less than equal to, to solve this we will divide the main inequality into two inequality and then get the upper and lower limits of x which will become the solution to the main inequality that is stated in the question.
So first we will divide the main inequality into two inequalities from which we get
\[\begin{align}
  & 2x-1\le 1......\left( 1 \right) \\
 & 2x-1\ge -23......\left( 2 \right) \\
\end{align}\]
Now we will first look at equation 1, to solve this inequality we will first add 1 on both sides of inequality and then we are going to divide both sides of inequality by 2 and so we get
$\begin{align}
  & \Rightarrow 2x-1+1\le 1+1 \\
 & \Rightarrow 2x\le 2 \\
 & \Rightarrow \dfrac{2x}{2}\le \dfrac{2}{2} \\
 & \Rightarrow x\le 1 \\
\end{align}$
So from equation 1 we get that the maximum value of x for which this inequality holds true is 1 so $x\le 1$ will become the upper limit of the interval of x
Now to find the lower limit of x we will be using equation 2, in this we are going to add 1 on both sides of inequality and then we will divide both sides of inequality by 2 and we will get
$\begin{align}
  & \Rightarrow 2x-1+1\ge -23+1 \\
 & \Rightarrow 2x\ge -22 \\
 & \Rightarrow \dfrac{2x}{2}\ge -\dfrac{22}{2} \\
 & \Rightarrow x\ge -11 \\
\end{align}$
So in this we get the minimum value x can have such that the inequality holds true as \[x\ge -11\]
So when we add both the results of equation 1 and 2 we will get the final interval of x as \[-11\le x\le 1\]
So the interval of x for which the given interval holds true is \[-11\le x\le 1\]


Note: In the above type of question the general mistakes that can happen is we forget to change the inequality sign while changing the sign of the variable, always remember that the inequality sign will change when there is a change in sign of the variable.