1. What is an Absolute Value Inequalities Calculator?

Definition: This calculator solves absolute value inequalities of the form \( a \cdot |bx + c| + d \lessgtr e \), where \( \lessgtr \) can be \( <, \leq, >, \geq \). The absolute value \( |x| \) represents the distance of \( x \) from zero, always non-negative.

Purpose: It helps students, engineers, and mathematicians solve inequalities involving absolute values, which are useful in applications like distance constraints, error analysis, and optimization.

2. How Does the Calculator Work?

The calculator solves the inequality \( a \cdot |bx + c| + d \lessgtr e \) using the following steps:

• Simplify: \( |bx + c| \lessgtr \frac{e - d}{a} \), flipping the sign if \( a < 0 \).
• If \( \frac{e - d}{a} < 0 \) and the inequality is \( < \) or \( \leq \), there are no solutions.
• If \( \frac{e - d}{a} < 0 \) and the inequality is \( > \) or \( \geq \), the solution is all real numbers.
• For \( < \) or \( \leq \): Solve \( -\frac{e - d}{a} < bx + c < \frac{e - d}{a} \).
• For \( > \) or \( \geq \): Solve \( bx + c < -\frac{e - d}{a} \) or \( bx + c > \frac{e - d}{a} \).

Where:

• \( a \): Multiplier of the absolute value expression;
• \( b \): Coefficient of \( x \) inside the absolute value;
• \( c \): Constant inside the absolute value;
• \( d \): Constant added outside the absolute value;
• \( e \): Right-hand side of the inequality;
• Results are displayed in interval notation with 4 decimal places or in scientific notation if less than 0.001.

Graphing Instructions

Step	Description
1	Plot \( y = bx + c \), a straight line through points \( (0, c) \) and \( (-\frac{c}{b}, 0) \).
2	Reflect the negative part (\( y < 0 \)) through the x-axis to get \( |bx + c| \).
3	Multiply by \( a \) to adjust slope; if \( a < 0 \), reflect through x-axis.
4	Add \( d \) to shift the plot vertically.
5	Draw \( y = e \); for \( < \), find where the V-shaped graph is below the line; for \( > \), find where it’s above.

Notes:

• The graph of \( a \cdot |bx + c| + d \) is V-shaped, with a vertex at \( x = -\frac{c}{b} \).
• Solutions are the intervals where the graph satisfies the inequality relative to \( y = e \).

Steps to Use:

• Enter coefficients \( a, b, c, d, e \).
• Select the inequality type (\( <, \leq, >, \geq \)).
• Click "Calculate" to compute solutions.
• View results in interval notation.

3. Importance of Absolute Value Inequalities

Solving absolute value inequalities is crucial for:

• Distance Constraints: Used in physics and engineering to model acceptable ranges for variables.
• Error Analysis: In statistics, to define bounds for deviations.
• Optimization: Helps set constraints in mathematical modeling and operations research.

4. Using the Calculator

Example 1: Solve \( -2 \cdot |x + 3| \geq 0 \):

• Inputs: \( a = -2, b = 1, c = 3, d = 0, e = 0 \), inequality \( \geq \);
• Simplify: \( |x + 3| \leq 0 \) (flip sign since \( a < 0 \));
• Since \( |x + 3| \geq 0 \), equality holds when \( x + 3 = 0 \Rightarrow x = -3 \);
• Result: \( [-3.0000, -3.0000] \).

Example 2: Solve \( |x - 8| + 7 \geq 6 \):

• Inputs: \( a = 1, b = 1, c = -8, d = 7, e = 6 \), inequality \( \geq \);
• Simplify: \( |x - 8| \geq -1 \);
• Since absolute value is always non-negative, the solution is all real numbers;
• Result: \( (-\infty, \infty) \).

Example 3: Solve \( 2 \cdot |x + 1| + 1 < 5 \):

• Inputs: \( a = 2, b = 1, c = 1, d = 1, e = 5 \), inequality \( < \);
• Simplify: \( |x + 1| < \frac{5 - 1}{2} = 2 \);
• Solve: \( -2 < x + 1 < 2 \Rightarrow -3 < x < 1 \);
• Result: \( (-3.0000, 1.0000) \).

5. Frequently Asked Questions (FAQ)

Q: What is an absolute value inequality?
A: It’s an inequality involving an absolute value expression, like \( |x| \lessgtr a \), representing distances on a number line.

Q: Why might an absolute value inequality have no solutions?
A: If \( \frac{e - d}{a} < 0 \) for a \( < \) or \( \leq \) inequality, the absolute value cannot satisfy the condition.

Q: How does the sign of \( a \) affect the solution?
A: If \( a < 0 \), the inequality sign flips (e.g., \( < \) becomes \( > \)), affecting the solution intervals.

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