Answer
Verified
423k+ 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
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The cell wall of prokaryotes are made up of a Cellulose class 9 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE
Select the word that is correctly spelled a Twelveth class 10 english CBSE
a Tabulate the differences in the characteristics of class 12 chemistry CBSE