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Rewrite the inequation $\left| x-1 \right|\le 3$ in the form $a\le x\le b$. Then the value of b – a =?

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Last updated date: 19th Jul 2024
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Answer
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Hint: If we are given an inequation in the form $\left| f\left( x \right) \right|\le k$ where f(x) is a function of x and k is a positive real number, then we can square both the sides of inequation. Since both the sides of the inequation are positive, squaring both the sides of the inequation will not affect the sign of the inequality. Using this, we can solve this question.

Complete step-by-step answer:

Before proceeding with the question, we must know the concept that will be required to solve this question.

If we are given an inequation that is written in the form $\left| f\left( x \right) \right|\le k$ where f(x) is a function of x and k is a positive real number, then to solve this inequation, we can square both the sides of this inequation. Squaring both the sides of this inequation will not affect the sign of inequality since both the sides of this inequation are positive.

In this question, we are given an inequation $\left| x-1 \right|\le 3$ and we have to convert this in the form $a\le x\le b$.

Since both the sides of the inequation $\left| x-1 \right|\le 3$ are positive, we can square both the sides. So, we get,
$\begin{align}
  & {{\left( \left| x-1 \right| \right)}^{2}}\le {{3}^{2}} \\
 & \Rightarrow {{\left( x-1 \right)}^{2}}\le {{3}^{2}} \\
\end{align}$

We have a formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$. Using this formula in the above equation, we get,
$\begin{align}
  & {{x}^{2}}-2x+1\le 9 \\
 & \Rightarrow {{x}^{2}}-2x+1-9\le 0 \\
 & \Rightarrow {{x}^{2}}-2x-8\le 0 \\
 & \Rightarrow {{x}^{2}}+2x-4x-8\le 0 \\
 & \Rightarrow x\left( x+2 \right)-4\left( x+2 \right)\le 0 \\
 & \Rightarrow \left( x-4 \right)\left( x+2 \right)\le 0 \\
 & \Rightarrow -2\le x\le 4 \\
\end{align}$

Comparing this with $a\le x\le b$, we get,
a = -2 and b = 4
So, the value of b – a = 4 – (-2) = 4 + 2 = 6.

Note: There is a possibility that in a hurry two solve this question, one may give the obtained inequation i.e. $-2\le x\le 4$ as the answer. But since we are required to find the value of b – a, we have to give b – a as the answer.