Problem 35 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. If there is no solution, write no solution. $$ |x|=-2 $$

Step-by-Step Solution

Answer

The equation has no solution as an absolute value cannot be negative.

Step 1: Understand Absolute Value

The absolute value of a number refers to its distance from zero on a number line. It is always non-negative. An absolute value can be represented as \(|x|\). Therefore, \(|x|\) can be any non-negative number.

Step 2: Solve the Equation

We attempt to solve the equation \(|x|=-2\). As stated in the first step, the absolute value of a number cannot be negative, hence it's impossible that \(|x|\) equals -2.

Step 3: Identify if the equation has a solution or not

Since the absolute value of a number cannot be a negative number and in this case it's defined as -2, there's no value of x, for which \(|x|\) will be -2. Thus, the given equation has no solution.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial in math as they help us denote values less than zero, commonly used in various fields like science and finance. When you see a number with a minus sign before it, like \(-5\), it's a negative number. This tells us it represents a position to the left of zero on a number line.

• Zero is not positive or negative; it's a neutral point.
• Numbers can only be negative if they are less than zero.

Negative numbers pair with positives to illustrate differences or changes in value. Although they are immensely helpful in calculations, negative values cause a challenge when dealing with certain mathematical concepts like absolute values. Since absolute values always return positive results, negative numbers cannot exist in absolute value solutions.

Equations

Equations are mathematical statements that express the equality between two expressions, employing an equal sign \(=\) to bridge them. An equation consists of variables, numbers, and operational symbols, and your task is to determine the values of these variables that satisfy the equality.For instance, with the equation \(x + 3 = 7\), we're tasked to find the value of \(x\) that makes this statement true. By simplifying, we find \(x = 4\). Equations can become more complex when introducing operations like multiplication, division, exponents, and roots. While most equations have solutions, certain types, such as those involving absolute values, can pose challenges. Knowing whether a solution exists or not requires analyzing the relationships within the equation, which leads us to the topic of solution existence.

Solution Existence

The existence of a solution in mathematical problems is about determining whether there is an answer that satisfies the equation or statement given. Understanding when a solution is possible is crucial, as seen in the problem \(|x| = -2\).In this equation, we have the absolute value of \(x\) set to -2. However, it's important to know that absolute values measure distance from zero, meaning they can never be negative. Therefore:

• Absolute values can only be zero or positive.
• This means \(|x| = -2\) is impossible.

No number's distance from zero is negative, confirming that this equation has no solution. Understanding when a problem has no solution saves time and avoids unnecessary calculations. It also deepens the comprehension of fundamental mathematical principles.