Question

The integral values of \(x\) satisfying the equation \(2^{110} x+10-2^{x}=\left|2^{x}-1\right|+1\) is: (b) \([1, \infty)\) (a) \((-\infty, 0)\) (c) \([0, \infty)\) (d) \(2 n: n \in I\)

Short answer

Based on the given step by step solution, the integral values of \(x\) that satisfy the equation are given by the set \(x = 2n\) for \(n \in \mathbb{Z}\).

Step-by-step solution

Rewrite the equation for clarity

To rewrite the equation, we first observe that the given equation is: \(2^{110}x + 10 - 2^x = |2^x - 1| + 1\)

Subtract 1 from both sides of the equation

Subtracting 1 from both sides of the equation, we obtain: \(2^{110}x + 9 - 2^x = |2^x - 1|\)

Analyze the possible cases for the absolute value function

Since we have an absolute value function in our equation, we need to consider the possible cases for the expression inside the absolute value function. There are two cases: 1. \(2^x \geq 1\): In this case, \(|2^x - 1| = 2^x - 1\) 2. \(2^x < 1\): In this case, \(|2^x - 1| = 1 - 2^x\) Let's analyze the equation for both cases.

Rewrite the equation for Case 1

For Case 1, when \(2^x \geq 1\), we have: \(2^{110}x + 9 - 2^x = 2^x - 1\) Now, move \(2^x\) to the other side of the equation: \(2^{110}x + 10 = 2^{x+1}\)

Rewrite the equation for Case 2

For Case 2, when \(2^x < 1\), we have: \(2^{110}x + 9 - 2^x = 1 - 2^x\) Now, move \(2^x\) to the other side of the equation: \(2^{110}x + 9 = 1\)

Analyzing the values of x from the rewritten equation in Case 1

From the rewritten equation in Case 1 (\(2^{110}x + 10 = 2^{x+1}\)), we can observe that it is true for all even integers as \(2^{110}\) is an even power of 2 and the presence of \(2^{x+1}\) on the other side. Thus, we can write this as: \(x = 2n, \quad \forall n \in \mathbb{Z}\).

Analyzing the values of x from the rewritten equation in Case 2

From the rewritten equation in Case 2 (\(2^{110}x + 9 = 1\)), we observe that there exists no value of x that makes an integer solution. Hence, there is no solution in this case.

Combining results for the solution

Finally, combining the results from steps 6 and 7, we conclude that the integral values of \(x\) satisfying the equation are given by the set \(x = 2n\) for \(n \in \mathbb{Z}\). This corresponds to option (d).

Key concepts

Absolute Value Equations

Absolute value equations relate to solving equations that involve the absolute value operation. An absolute value, denoted by \(|x|\), represents the distance of a number from zero on the number line.
It always results in a non-negative number, regardless of whether the input is positive or negative.

To solve an equation with an absolute value, you need to consider two possible cases:

• One case is when the expression inside the absolute value is non-negative.
• Another case is when the expression inside the absolute value is negative. For a negative inside value, take the negative of the entire expression to eliminate the absolute value.

For example, in the equation \(|2^x - 1| = y\), you consider:

• If \(2^x \ge 1\), then \(2^x - 1 = y\).
• If \(2^x < 1\), then \(1 - 2^x = y\).

Using these cases, you create different equations to solve. Always check which case applies before finalizing the solution to ensure it fits the given conditions of the absolute value expression.

Even Integers

Even integers are numbers that can be divided by the number 2 without leaving a remainder.
These numbers are formulaically represented as \(2n\), where \(n\) is any integer.

In problems where you need to find solutions in terms of even integers, it's important to recognize patterns or conditions where expressions equal or simplify to forms like \(2m\).

For instance, in the exercise, the solution shows:

• The equation modifies such that, due to the structure involving powers of 2, \(x\) turns out to be an even integer, written as \(2n\).
• This indicates that for any integer \(n\), \(x = 2n\) satisfies the condition, hence belongs to the set of all even integers.

Recognizing this structure simplifies finding possible solutions and ruling out incompatible values. This also highlights the importance of the relationships between variables when deducing integer properties.

Exponential Equations

Exponential equations involve variables in the exponent. They often take the form \(a^{x} = b\), where solving for \(x\) requires understanding the properties of exponents.
Approaching these equations involves:

• Evaluating cases based on the sign and size of the base or other terms compared to it.
• Utilization of logarithmic functions to isolate variables when straightforward simplification isn't feasible.

When dissecting the equation \(2^{110}x + 10 - 2^x = |2^x - 1| + 1\), we see:

• The presence of \(2^{x}\) suggests leverages of exponential and logarithmic rules might be necessary.
• Breaking down the equation involves both exponential manipulation and understanding the behavior of large exponents.

Combining exponential rules with properties of absolute values and integer distinctions helps solve such complex equations effectively. Always consider the number behavior in powers to simplify and derive possible solutions efficiently.