Answer
Verified
477.9k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
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
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE