Absolute Value Inequality

Definition

An absolute value inequality is a mathematical expression that involves the absolute value of a variable or expression being compared to a constant value using an inequality symbol such as greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). The absolute value function represents the distance of a number from zero on the number line, and the inequality compares this distance to a specified value.

5 Must Know Facts For Your Next Test

1. Absolute value inequalities can be used to model real-world situations where a quantity must be within a certain range, such as temperature, distance, or time.
2. Solving absolute value inequalities involves isolating the absolute value expression and then considering the two possible cases: when the expression is greater than or equal to the constant, and when the expression is less than or equal to the constant.
3. The solution set for an absolute value inequality can be represented on a number line, with the points where the inequality is true shown as an interval or a union of intervals.
4. Absolute value inequalities can be combined with other types of inequalities, such as linear or quadratic inequalities, to create more complex mathematical models.
5. The graphical representation of an absolute value inequality is a V-shaped region on the coordinate plane, with the vertex of the V corresponding to the point where the absolute value expression is equal to the constant.

Review Questions

• Explain how the concept of absolute value relates to solving linear inequalities involving absolute value expressions.
  • When solving a linear inequality with an absolute value expression, such as $|x - 3| \geq 5$, the absolute value represents the distance of the variable $x$ from the constant 3 on the number line. To solve this inequality, we need to consider two cases: when $x - 3 \geq 5$ (which gives $x \geq 8$) and when $x - 3 \leq -5$ (which gives $x \leq -2$). The solution set is the union of these two intervals, which represents all the values of $x$ that satisfy the inequality.
• Describe the process of solving a quadratic inequality that involves an absolute value expression, such as $|x^2 - 4x + 3| \leq 2$.
  • To solve a quadratic inequality with an absolute value expression, we first need to isolate the absolute value term. In this case, we can rewrite the inequality as $-2 \leq x^2 - 4x + 3 \leq 2$. Next, we can solve the two resulting linear inequalities separately: $x^2 - 4x + 3 \geq -2$ and $x^2 - 4x + 3 \leq 2$. Solving these inequalities will give us the two intervals that make up the solution set. The final step is to find the intersection of these two intervals, which represents all the values of $x$ that satisfy the original quadratic inequality involving the absolute value expression.
• Analyze how the graphical representation of an absolute value inequality can be used to understand the solution set and make connections to the algebraic solution process.
  • The graph of an absolute value inequality is a V-shaped region on the coordinate plane, with the vertex of the V corresponding to the point where the absolute value expression is equal to the constant. To solve the inequality, we need to consider the two half-planes created by the V-shaped region. The solution set is the union of the two intervals where the inequality is true. By visualizing the graph, we can gain insights into the solution process, such as understanding how the distance of the variable from the constant affects the solution, and how the shape of the V-shaped region changes based on the type of inequality (e.g., greater than, less than, or equal to). The graphical representation can also help us make connections between the algebraic and geometric aspects of solving absolute value inequalities.

Related terms

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of whether the number is positive or negative.

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.

Quadratic Inequality: A quadratic inequality is an inequality that involves a quadratic expression, such as $ax^2 + bx + c \geq 0$, where $a$, $b$, and $c$ are constants.