Problem 35 - Precalculus

Question

Rewrite each expression without using absolute value notation. $$|x-3|+|x-4| \text { given that } 3

Step-by-Step Solution

Answer

The expression simplifies to 1 for \(3 < x < 4\).

Step 1: Understand Absolute Value without Notation

The expression involves two absolute values: \(|x-3|\) and \(|x-4|\). Absolute value \(|a|\) represents the distance of \(a\) from zero on the number line. Thus, \(|a| = a\) for \(a \geq 0\) and \(|a| = -a\) for \(a < 0\). Here, we need to rewrite the expression without absolute values given the condition \(3 < x < 4\).

Step 2: Evaluate |x-3| Condition

Since \(3 < x\), \(x - 3 > 0\). Thus, \(|x - 3| = x - 3\) because in this range, \(x-3\) is positive.

Step 3: Evaluate |x-4| Condition

Since \(x < 4\), \(x - 4 < 0\). Hence, \(|x - 4| = -(x - 4) = 4 - x\), as in this range, \(x-4\) is negative.

Step 4: Combine Results

Combine the results from the previous steps: Substitute \(|x - 3| = x - 3\) and \(|x - 4| = 4 - x\) into the original expression. This gives us: \[(x - 3) + (4 - x)\].

Step 5: Simplify the Expression

Simplify the expression \((x - 3) + (4 - x)\):- The \(x\) terms cancel out, leaving \(-3 + 4\). - Simplifying further, we are left with \(1\).

Key Concepts

Understanding Number Lines

Number lines are straightforward, yet mighty tools in mathematics. They're essentially a straight line where each point corresponds to a number. The center point is zero, and numbers increase as you move to the right, while they decrease as you move to the left. When dealing with absolute values, number lines help us visually grasp distances between numbers.

Let's say we have a number 3 on a number line. If we move towards right, increasing the value, any number to the right of 3 (for example 4 or 5) is greater than 3. Similarly, any number to the left, such as 2 or 1, is less than 3. This visualization assists us when working with absolute values, as absolute value symbolizes distance on the number line. So, \(|x-3|\) signifies how far \(x\) is from 3, regardless of direction.

• If \(x\) is to the right of 3, \(|x-3| = x-3\). It's as if subtracting 3 from \(x\) gives you that exact distance.
• If \(x\) is to the left of 3, \(|x-3| = 3-x\), flipping the sign to ensure a positive distance.

The Beauty of Piecewise Functions

Piecewise functions in math allow us to define different expressions or equations over various intervals. This flexibility is key in effectively characterizing situations that differ based on the input.

For instance, consider the expression \(|x-3| + |x-4|\), under the condition \(3

Between 3 and 4, the direction or sign of the expression matters, adjusting according to which part of the line you're evaluating.

Piecewise functions have breaks at certain points, just like where the condition changes in this example.

The beauty of piecewise functions is their power in capturing real-world phenomena by accounting for different scenarios and their effects.

Decoding Algebraic Expressions

Algebraic expressions are key components in math, composed of variables, numbers, and operations such as addition or subtraction. Understanding them, especially in the context of absolute values, is essential.

Breaking down an expression like this one: \((x - 3) + (4 - x)\), we can see how algebraic skills help simplify it. By exploring how each term acts, we arrive at the simplest form. For example, note how the \(x\) and \(-x\) terms cancel each other out, emphasizing the clarity algebra provides by stripping away unnecessary complexity.

• The 'hidden' negative sign came from changing \(|x-4| = 4 - x\), impacting the final results even though it's easy to overlook.
• Simplifying these expressions enriches our understanding of the relationships between terms and their impact on equations.
• Moreover, algebraic expressions like these form fundamental building blocks for more complex equations and functions encountered later in math studies.