Question

If \(|x|+x=4\), then which one of the following is odd? (A) \(x^2+3 x\) (B) \(x^2+3 x+2\) (C) \(x^2+4 x\) (D) \(x^2+4 x+2\) (E) \(x^2+4 x+3\)

Short answer

The correct answer is (E) \(x^2 + 4x + 3\).

Step-by-step solution

Understanding Absolute Value Equation

The equation is given as \(|x| + x = 4\). We have to consider cases for \(x\): when \(x\) is non-negative (\(x \geq 0\)) and when \(x\) is negative (\(x < 0\)).

Case Analysis on "x"

* If \(x \geq 0\), then \(|x| = x\). Therefore, the equation becomes:\[ x + x = 4 \Rightarrow 2x = 4 \Rightarrow x = 2 \].* If \(x < 0\), then \(|x| = -x\). Therefore, the equation becomes:\[ -x + x = 4 \Rightarrow 0 = 4 \], which is a contradiction. Thus, \(x < 0\) cannot be true.

Find "x" Value

From Step 2, we determined that \(x = 2\) is the solution.

Checking Each Option for Odd Result

Substitute \(x = 2\) into each option to determine which expression is odd:A. \(x^2 + 3x = 2^2 + 3 \cdot 2 = 4 + 6 = 10\) (Even)B. \(x^2 + 3x + 2 = 2^2 + 3 \cdot 2 + 2 = 10 + 2 = 12\) (Even)C. \(x^2 + 4x = 2^2 + 4 \cdot 2 = 4 + 8 = 12\) (Even)D. \(x^2 + 4x + 2 = 2^2 + 4 \cdot 2 + 2 = 12 + 2 = 14\) (Even)E. \(x^2 + 4x + 3 = 2^2 + 4 \cdot 2 + 3 = 12 + 3 = 15\) (Odd)

Determine the Correct Answer

Only option E results in an odd number when \(x = 2\), therefore the correct answer is (E): \(x^2 + 4x + 3\).

Key concepts

Case Analysis in Absolute Value Equations

Absolute value equations often require us to consider different cases since the absolute value affects the equation depending on whether the variable is positive or negative. For the equation \(|x| + x = 4\), we need to analyze the two main cases where the behavior of \(|x|\) changes based on the value of \(x\).
Let's break it down into two scenarios:

• **When \(x \geq 0\):** Here, the absolute value of \(x\) is simply \(x\) itself, meaning the equation becomes \(x + x = 4\). Solving, we get \(2x = 4\), and hence \(x = 2\).
• **When \(x < 0\):** In this instance, \(|x| = -x\), so the equation \(-x + x = 4\) simplifies to \(0 = 4\), which is not possible. Therefore, this case leads to a contradiction.

By considering these two cases separately, we can find valid solutions. For \(|x| + x = 4\), the only solution from our analysis is \(x = 2\) since \(-x + x = 4\) doesn't work.
Understanding case analysis helps in solving problems systematically when absolute values are involved.

Basics of Solving Equations

Solving equations is all about finding the value of the variable that makes the equation true. In our example, we started with the equation \(|x| + x = 4\). The goal was to isolate the variable, \(x\), and determine its value using logical steps.
When solving an equation:

• Identify any special conditions on the variable, such as conditions induced by absolute value or other constraints.
• Break down the problem, as we did using case analysis to handle \(|x|\), and consider each possibility logically.
• Solve the simplified equations by performing basic operations like addition, subtraction, multiplication, or division to isolate the variable on one side.

In our problem, determining \(x = 2\) meant substituting this into each provided option. This substitution allowed us to verify which expressions resulted in odd or even outcomes. Techniques used here give a foundation for solving a wide variety of equations.

Understanding Odd and Even Numbers

When working with numbers, distinguishing between odd and even is important. An even number is any number that can be divided by 2 without leaving a remainder, whereas an odd number leaves a remainder of 1 when divided by 2.
In our context, once we found \(x = 2\), we substituted it into the given options to determine their parity:

• Calculate the expression.
• Determine if the result is odd or even by checking divisibility by 2.

For instance:- Option E: \(x^2 + 4x + 3 = 15\), and since 15 divided by 2 gives a remainder of 1, it's odd.
Understanding these concepts ensures accuracy in tasks where you might need to differentiate between results based on their parity. Knowing how to test if a number is odd or even is a fundamental skill in mathematics.