Which of the following has exactly one solution ?

It is. Both x = 1 and x = 2 satisfy 2|x| = 2^x.

Can you please post a detailed solution for this.

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is there a specific way to solve this question apart from trial and error?

\( C. 2|x| = 2^x \)
case (i) x is -ve, then
\( |x| = x --> 2*x = \frac{1}{2^{x}} \), which is not possible for any x
case (ii) x is +ve, then
\( |x| = x --> 2*x = 2^x \)
above equation holds good for x = 1 (2*1 = 2^1) and x = 2 (2*2 = 2^2), so option C has two solutions for x, so not correct

E. |x+2| = -x
case (i) x is -ve, then take
\( x = -1 --> |-1+2| = -(-1) \)
\( 1 = 1 \), so x takes -1 ( one solution)
\( x= -2 --> |-2+2| = -(-2) \)
\( 0 = 2 \), so x can't take -2
\( x = -3 --> |-3+2| = -(-3) \) so x can't take -3, and so for other negative numbers

case (ii) x is +ve, then
\( |x+2| = -x \) becomes x+2 = -x --> x = -1
so, x can take -1, from case (i) & (ii), x has only one solution (-1)

so option E is correct

Solution:

We can solve the equations starting with D since D is the easiest equation of the ones given.

D. 2|x| = 2x - 1

If x is positive, we have 2x = 2x - 1, which yields no solution.

If x is negative, we have -2x = 2x - 1. Solving it, we get:

-4x = -1

x = -1/-4 = 1/4

Since we assumed x was negative, the positive value x = 1/4 may not satisfy the equation. Indeed, if we substitute x = 1/4, we get:

2|1/4| ≟ 2(1/4) - 1

2 * 1/4 ≟ 1/2 - 1

1/2 ≟ -1/2

Since 1/2 is not equal to -1/2, x = 1/4 is not a solution for 2|x| = 2x - 1. Thus, this equation has no solutions.

Next, let’s solve the equation in answer choice E since it is similarly easy to solve.

E. |x + 2| = -x

If (x + 2) is positive, then:

x + 2 = -x

2x = -2

x = -1

We can verify that x = -1 is indeed a solution for |x + 2| = -x.

If (x + 2) is negative, then:

-x - 2 = -x

x + 2 = x

No value of x will satisfy the equation above. Hence, the equation |x + 2| = -x has only one solution.

Answer: E

Is this an official question? The rule is that absolute value cannot give us a negative result but here we take it as the correct answer. How can |1| be equal to -1?

|x + 2| = −x has one solution x = -1:

|-1 + 2| = −(-1);

|1| = 1;

1 = 1.

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