Problem 13 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. If there is no solution, write no solution. $$ |x|=5.5 $$

Step-by-Step Solution

Answer

The solutions to the equation \(|x| = 5.5\) are \(x = 5.5\) and \(x = -5.5\).

Step 1: Understanding Absolute Value

An absolute value of a number is its distance from zero. Therefore, the absolute value of a number is always nonnegative. For \(|x| = a\), there could be two possible solutions \(x = a\) and \(x = -a\).

Step 2: Apply Absolute Value Rule

Applying the rule to the equation \(|x| = 5.5\), we have two possible solutions. First, \(x = 5.5\) and second, \(x = -5.5\).

Key Concepts

Understanding Absolute Value

When we talk about the absolute value of a number, we are referring to its distance from zero on the number line, without considering which direction from zero it is. This distance is always a positive value or zero because absolute value represents magnitude without regard to sign. In mathematical terms, for any real number 'a', the absolute value is denoted as \(|a|\), which reads as 'the absolute value of a'.

Here's a simple way to visualize it: if you walked 5 steps to the right of zero, you'd be at +5, but if you walked the same distance to the left, you'd be at -5. Regardless of the direction, your distance from the starting point (zero) is the same: 5 steps. This is what absolute value measures—the magnitude or size of that distance without any sign associated with it, making \(|-5| = 5\) as well as \(|5| = 5\).

Understanding this concept is critical when you're working with absolute value equations because you'll often encounter two potential solutions, and knowing that absolute value represents a non-negative result will guide you to accurate answers.

Solving Absolute Value Equations

To solve absolute value equations, it's essential to recall that such equations can have two possible solutions. This is due to the nature of absolute value—being the distance from zero—allowing for both a positive and a negative value to have the same absolute value.

Let's take the given equation, \(|x| = 5.5\). Since absolute value refers to the distance from zero, there are two numbers that are exactly 5.5 units away from zero: +5.5 and -5.5. Consequently, in order to solve this equation, you split it into two separate equations:
1. \(x = 5.5\)
2. \(x = -5.5\)

By finding both the positive and negative values that satisfy the original equation, you cover all possible solutions. If instead the equation had a negative value (which is not possible in this context), it would have no solution, since absolute values are inherently nonnegative. Remember, it is the distance away from zero and distances cannot be negative.

Mental Math Strategies

Solving absolute value equations with mental math requires you to quickly assess the equation and apply a few key strategies. Here are some points that can help:

• Identify the Absolute Value: Recognize the absolute value symbol and remember that it signifies the distance from zero, which is always nonnegative.
• Split the Equation: Conceptualize that you need to consider both the positive and the negative number that correspond to that distance.
• Simplify the Problem: In the case of our example \(|x| = 5.5\), think of the number as how far you are from zero and simply note that there are two positions you can be, 5.5 units to the right or left of zero, leading to the quick recognition of both solutions: +5.5 and -5.5.
• Estimation: When working with more complicated expressions, estimate the results by rounding to nearby numbers that are easier to work with mentally.
• Pattern Recognition: Over time, familiarize yourself with common patterns and results of absolute value equations to speed up the process.

Through regular practice and the application of these strategies, you can improve your ability to solve absolute value equations using mental math, making it a handy skill in various mathematical contexts.