Problem 2 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(|2 x-1|=10\) is an absolute value _____ and \(|2 x-1|>10\) is an absolute value _____.

Step-by-Step Solution

Answer

equation, inequality.

Step 1: Identify the Equation

Recognize that the expression given is \(|2x - 1| = 10\). This is an absolute value equation.

Step 2: Define the Inequality

Recognize that the expression given is \(|2x - 1| > 10\). This is an absolute value inequality.

Key Concepts

Absolute Value Inequality

Understanding absolute value inequalities can be incredibly useful in problem-solving. An absolute value inequality compares the size of an algebraic expression within an absolute value set against a constant. For example, in the expression \( |2x - 1| > 10 \), the absolute value \( |2x - 1| \) is greater than 10. This means the distance from zero on a number line must be more than 10.

To solve such inequalities, it's crucial to split the problem into two parts:

• The expression itself is greater than the constant, \( 2x - 1 > 10 \).
• The expression is less than the negative of the constant, \( 2x - 1 < -10 \).

Combining these solutions provides us with all possible values of \( x \) that satisfy the inequality. These intervals reflect the value ranges for which the expression exceeds the specified threshold set by the inequality.

Solving Absolute Value Equations

Solving absolute value equations can seem complex, but by breaking down the steps, it becomes manageable. Absolute value equations, like \( |2x - 1| = 10 \), represent two different scenarios because the absolute value takes the distance from zero, meaning both positive and negative solutions may exist.
When solving \( |2x - 1| = 10 \):

• You consider the scenario where \( 2x - 1 = 10 \), which provides a positive root.
• Also, consider \( 2x - 1 = -10 \), which reflects the negative root.

By solving these two equations, you obtain the specific values of \( x \) that solve the absolute value equation. This double-sided approach ensures that all possible solutions are considered, covering both positive and negative scenarios.

Algebraic Expressions

Algebraic expressions form the backbone of solving mathematical problems, especially when dealing with absolute values. An algebraic expression like \( 2x - 1 \) consists of variables, constants, and operators combined together to represent a specific value.
In the context of absolute values, understanding how to manipulate and simplify these expressions is key to effectively solving equations and inequalities.

• Variables represent unknown quantities and can vary.
• Constants are fixed values that the expressions are often compared against.
• Operators like addition, subtraction, multiplication, and division help in manipulating the values within the expressions.

Managing algebraic expressions allows us to transform and rearrange them in ways that simplify the process of solving equations, whether they include direct calculations or more complex absolute value situations.