Problem 36 - Precalculus

Question

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

Step-by-Step Solution

Answer

1

Step 1: Understand Absolute Value

Absolute value represents the distance from zero on the number line. Thus, \(|a| = a\) if \(a \geq 0\) and \(|a| = -a\) if \(a < 0\). When evaluating an expression, consider the sign of the expression inside the absolute value.

Step 2: Evaluate each Absolute Value for x=4

First, evaluate \(|x - 3|\) for \(x=4\):\[ |x - 3| = |4 - 3| = |1| = 1 \]Next, evaluate \(|x - 4|\) for \(x=4\): \[ |x - 4| = |4 - 4| = |0| = 0 \]

Step 3: Rewrite Expression Without Absolute Value

Substitute the evaluated expressions into the original equation:\( |x - 3| + |x - 4| \)\[ = 1 + 0 \]Thus, rewrite the expression as \(1\).

Key Concepts

Distance on the Number Line

The concept of distance on the number line is essential for understanding absolute value. Imagine a straight line with numbers placed at equal intervals, where zero is typically the center point. The distance a number is from zero is what we call absolute value. For any real number, the absolute value is always positive because distance cannot be negative. When we say the absolute value of a number, like \(|x-3|\), we mean how far \(x-3\) is from zero.

• If \(x-3\) is positive or zero, then the distance is equal to \(x-3\) itself.
• If \(x-3\) is negative, its distance is \(-(x-3)\).

By understanding that absolute value is about distance, we can visually imagine when numbers change direction on the number line, or where numbers exist relative to each other.

Evaluate Expressions

Evaluating expressions means finding their numerical value. When we deal with absolute values, it’s important to determine the value of the expression inside them first. Let's take our exercise as an example. We had to evaluate \(|x-3|\) and \(|x-4|\) for the specified value of \(x=4\).

• For \(|x-3|\), we substitute \(x=4\) to get \(|4-3|\), which simplifies to \(|1|\).
• Similarly, for \(|x-4|\), substitute \(x=4\) to find \(|4-4|\), simplifying it to \(|0|\).

After determining each absolute value, you replace them in the expression to simplify further. Recognizing the expression's sign before evaluating is key to simplifying correctly.

Substitute Expressions

Substitute expressions involve replacing variables or parts of expressions with their evaluated values. This can reduce an equation to its simplest form or solve it entirely. In our case, substituting values helps rewrite absolute values without using their notation.
Given our step-by-step solution:

• We first evaluated each expression: \(|x-3| = 1\) and \(|x-4| = 0\).
• After evaluating, the next step is to substitute these numbers back into the original equation: \(|x-3| + |x-4| = 1 + 0\).

This process of substitution helps translate an abstract math problem into an exact answer. Always simplify fully and check your work for accuracy, ensuring all values have been correctly replaced and added.