Problem 33 - Precalculus

Question

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

Step-by-Step Solution

Answer

-2x + 7

Step 1: Understand the Absolute Values

The given expression involves absolute values: \(|x-3|+|x-4|\). Absolute values simplify to linear expressions depending on whether the expression inside the absolute value is positive or negative.

Step 2: Consider the Condition x<3

Given that \(x < 3\), both \(x-3\) and \(x-4\) are negative. Hence, the absolute values can be rewritten using the property \(|a| = -a\) if \(a < 0\).

Step 3: Remove the Absolute Value for |x-3|

Since \(x<3\), the expression \(x-3\) is negative. Thus, \(|x-3| = -(x-3)\). Simplifying this gives \(|x-3| = -x + 3\).

Step 4: Remove the Absolute Value for |x-4|

Similarly, since \(x<3\), \(x-4\) is negative. Thus, \(|x-4| = -(x-4)\). Simplifying this gives \(|x-4| = -x + 4\).

Step 5: Combine the Simplified Expressions

Add the simplified expressions from steps 3 and 4: \(-x + 3 + (-x + 4)\) simplifies to \(-2x + 7\).

Key Concepts

Absolute Value Properties

Absolute value is a number's distance from zero on the number line, without considering direction. This means it is always non-negative. For example, the absolute value of both \(5\) and \(-5\) is \(5\). The mathematical sentence for absolute value is denoted by vertical bars, like \(|a|\).
Consider \(|x-3|\) in our exercise. Here, the concept used is that if \(a < 0\), then \(|a| = -a\). This is because when a number inside the absolute value is negative, its absolute form is its negative counterpart, which makes it positive.
This property allows us to remove the absolute value signs by using conditions given in the problem, like \(x < 3\) for the expression \(|x-3| + |x-4|\). Because both \(x-3\) and \(x-4\) are negative in this range, we apply the property to rewrite them without absolute value brackets.

Linear Expressions

Linear expressions are algebraic expressions that involve constant terms and terms with variables raised only to the first power. They are written in the form \(ax + b\), where \(a\) and \(b\) are constants.
In the context of our problem, \(|x-3|\) becomes \(-x+3\) and \(|x-4|\) becomes \(-x+4\) when simplified. These are both linear expressions because they follow the form \(ax + b\).
Understanding linear expressions is key in algebra because they form the basis for solving equations and inequalities. They represent lines on a graph, and in this case, they are used to evaluate values under specific conditions, such as when \(x < 3\). Remember, a linear expression changes predictably, and understanding that change is vital for solving many beginner algebra problems.

Simplifying Expressions

Simplifying expressions means making them as concise and clear as possible. It involves combining like terms and removing any unnecessary components, such as absolute value signs in this exercise.
For example, when we look at \(-x+3\) and \(-x+4\), we see that both have \(-x\) as a common term. By combining these like terms, we simplify the expression.
So, instead of dealing with two separate linear expressions, these can be combined to give \(-2x+7\). Remember: simplification does not change the value the expression represents; it only makes it easier to work with and understand. This step is crucial for solving more complex equations efficiently.

Inequalities

Inequalities express a relationship of one expression being greater than or less than another. They are symbolized by signs such as \(<\), \(>\), \(\leq\), and \(\geq\). Unlike equations, inequalities do not always represent definite values but rather a range of possible values.
In this exercise, we deal with the inequality \(x < 3\). This tells us the range of values \(x\) can take. It indicates that any number less than \(3\) satisfies the given condition, which impacts how we simplify the absolute value expressions.
Understanding how inequalities operate is critical for finding solutions that must fit within specific boundaries, like those found in real-world scenarios or complex math problems. Recognizing when and how to apply them helps solve problems involving conditions, like those given in many algebraic expressions and constraints.