If and , then the value of

-2 Explanation: 1. Analyze the Absolute Value Terms Given the condition : For : Since , then . Therefore, . For : Since , then (which means ). Since , the term is negative. Therefore, . 2. Substitute back into the Equation The original equation is: Substituting the absolute value expressions: 3. Test the Options Since this is a multiple-choice question and we have a specific constraint ( ), we can test the options that satisfy the constraint. Only option (a) satisfies . Let's check : Left Hand Side (LHS): Right Hand Side (RHS): Wait, let's re-examine the equation from the image image_741606.png carefully. The equation in the image is . Corrected Substitutions for : Since the bases are the same, we equate the exponents: Conclusion: The value of is , which satisfies the initial condition . Final Answer: The

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NIMCET 2009 — Mathematics PYQ

NIMCET | Mathematics | 2009

If $x < -1$ and $2^{∣ x + 1∣} - 2 x = ∣ 2^{x} - 1∣ + 1$ , then the value of $x$

Choose the correct answer:

Explanation

1. Analyze the Absolute Value Terms

Given the condition $x < -1$ :

For $|x+1|$ : Since $x < -1$ , then $x + 1 < 0$ . Therefore, $|x+1| = -(x+1)$ .

For $|2^x - 1|$ : Since $x < -1$ , then $2^x < 2^{-1}$ (which means $2^x < 0.5$ ). Since $0.5 < 1$ , the term $2^x - 1$ is negative. Therefore, $|2^x - 1| = -(2^x - 1) = 1 - 2^x$ .

2. Substitute back into the Equation

The original equation is:

$2^{|x+1|} - 2x = |2^x - 1| + 1$

Substituting the absolute value expressions:

$2^{-(x+1)} - 2x = (1 - 2^x) + 1$

$2^{-(x+1)} - 2x = 2 - 2^x$

3. Test the Options

Since this is a multiple-choice question and we have a specific constraint ( $x < -1$ ), we can test the options that satisfy the constraint. Only option (a) $x = -2$ satisfies $x < -1$ .

Let's check $x = -2$ :

Left Hand Side (LHS):

$2^{-(-2+1)} - 2(-2)$

$2^{-(-1)} + 4$

$2^1 + 4 = 6$

Right Hand Side (RHS):

$2 - 2^{-2}$

$2 - \frac{1}{4} = 1.75$

Wait, let's re-examine the equation from the image image_741606.png carefully. The equation in the image is $2^{|x+1|} - 2^x = |2^x - 1| + 1$ .

Corrected Substitutions for $2^{|x+1|} - 2^x = |2^x - 1| + 1$ :

$2^{-(x+1)} - 2^x = (1 - 2^x) + 1$

$2^{-(x+1)} - 2^x = 2 - 2^x$

$2^{-(x+1)} = 2$

Since the bases are the same, we equate the exponents:

$-(x + 1) = 1$

$-x - 1 = 1$

$-x = 2$

$x = -2$

Conclusion:

The value of $x$ is $-2$ , which satisfies the initial condition $x < -1$ .

Final Answer:

The correct option is (a) $-2$ .