Match each equation or inequality in Column I with the graph ofits solution set in Column II. | x | = -7

Verified step by step guidance

1. Step 1: Understand the absolute value concept. The absolute value of a number, denoted as |x|, represents its distance from zero on the number line, which is always non-negative.
2. Step 2: Analyze the given equation |x| = -7. Since absolute values are always non-negative, they cannot equal a negative number.
3. Step 3: Conclude that there is no real number x that satisfies the equation |x| = -7.
4. Step 4: Recognize that the solution set for this equation is the empty set, meaning there are no solutions.
5. Step 5: Match this understanding with the graph in Column II that represents an empty solution set, typically shown as no points on the number line.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For example, |3| = 3 and |-3| = 3. In the context of equations, |x| = a means x can be either a or -a, provided a is non-negative.

No Solution in Absolute Value Equations

An absolute value equation like |x| = -7 has no solution because the absolute value cannot be negative. The definition of absolute value ensures that it is always zero or positive, thus making it impossible for |x| to equal a negative number. This concept is crucial for understanding the limitations of absolute value equations.

Graphing Absolute Value Functions

The graph of an absolute value function, such as y = |x|, forms a 'V' shape, opening upwards. The vertex of the graph is at the origin (0,0), and it reflects symmetrically across the y-axis. Understanding how to graph these functions helps visualize the solution sets of absolute value equations and inequalities.