Question

If $|x-5|+|x-10|⩽20$, then x belongs to

• A. [2.5, 17.5]
• B. [-2.5, 17.5]
• C. [-7.5, 2.5]
• D. [-17.5, 2.5]

Answer 1

The correct option is B

[-2.5, 17.5]

To solve this, we want to remove the modulus.
$\begin{array}{ccc}|x-5|=x-5& if& x>5\\ =5-x& if& x<5\\ |x-10|=x-10& if& x>10\\ =10-x& if& x<10\end{array}Letf\left(x\right)=|x-5|+|x-10|ifx⩾10f\left(x\right)=x-5+x-10=2x-15f\left(x\right)⩽20\Rightarrow 2x-15⩽20x⩽17.5$
But we have already assumed that $x\ge 10$. so the interval will be the intersection of $x\ge 10$ and $x⩽$ 17.5, which is [10,17,5]
if x<5
f(x) = 5-x +10-x
= 15-2x

So 15-2x less than or equal to 20
i.e. x is greater than or equal to -2.5
But we have already assumed that x<5. So x belongs to [-2.5,5)

For 5 <x<10

f(x) = x-5+10-x
= 5 which is always less than 20
So for 5 <x<10 the inequality always holds true for this range.

now for these 3 cases if we take the union we get x belongs to [-2.5,17.5]