Absolute Value Inequality

Definition

An absolute value inequality is a mathematical expression that involves the absolute value of a variable or expression. It is used to represent a range of values that satisfy the inequality, where the absolute value of the variable or expression must be less than, greater than, or equal to a specified value.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities can be used to represent constraints or boundaries in real-world applications, such as in finance, engineering, and physics.
2. The solution set of an absolute value inequality is the set of all values of the variable that satisfy the inequality.
3. Absolute value inequalities can be solved using various methods, including graphing, algebraic manipulation, and the use of the properties of absolute value.
4. Absolute value inequalities can be classified into two main types: one-sided inequalities (e.g., $|x| \geq a$) and two-sided inequalities (e.g., $a \leq |x| \leq b$).
5. Solving absolute value inequalities often involves considering cases based on the sign of the expression inside the absolute value.

Review Questions

• Explain the connection between absolute value inequalities and linear inequalities.
  • Absolute value inequalities are a generalization of linear inequalities, as they can be rewritten in terms of linear inequalities. For example, the absolute value inequality $|x - 3| < 5$ can be expressed as the compound inequality $-5 < x - 3 < 5$, which involves two linear inequalities connected by the word 'and'. Understanding the relationship between absolute value inequalities and linear inequalities is crucial for solving problems involving these types of expressions.
• Describe the process of solving a two-sided absolute value inequality, such as $a \leq |x| \leq b$.
  • To solve a two-sided absolute value inequality, such as $a \leq |x| \leq b$, you would first consider the two cases: when $x \geq 0$ and when $x < 0$. For the case where $x \geq 0$, you can rewrite the inequality as $a \leq x \leq b$. For the case where $x < 0$, you can rewrite the inequality as $-b \leq x \leq -a$. The solution set is the intersection of these two cases, which represents the range of values that satisfy the original two-sided absolute value inequality.
• Analyze the impact of the signs of the constants in an absolute value inequality, such as $-3 \leq |x + 2| \leq 5$, on the solution set.
  • The signs of the constants in an absolute value inequality, such as $-3 \leq |x + 2| \leq 5$, have a significant impact on the solution set. In this example, the negative sign on the left-hand side inequality indicates that the solution set includes values of $x$ that are both greater than and less than $-2$, as long as the absolute value of $(x + 2)$ is greater than or equal to $3$. The positive sign on the right-hand side inequality means that the absolute value of $(x + 2)$ must be less than or equal to $5$. Analyzing the impact of the signs of the constants is crucial for correctly identifying the solution set of an absolute value inequality.

Related terms

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of its sign. It is denoted by vertical bars, such as |x|.

Linear Inequality: A linear inequality is an inequality that involves a linear expression, such as $ax + b \geq c$, where $a$, $b$, and $c$ are constants.

Compound Inequality: A compound inequality is a combination of two or more inequalities, connected by the words 'and' or 'or', that must be satisfied simultaneously.