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Absolute value equations

Absolute value equations

Absolute value equations are mathematical equations in which the variable is inside an absolute value symbol. The absolute value of a number represents its distance from zero on the number line, regardless of direction. An equation of the form ∣x∣=a|x| = a will have two potential solutions: x=ax = a and x=−ax = -a (where aa is a non-negative number).

To solve absolute value equations, you can set up separate cases based on the definition of absolute value:

  1. For ∣x∣=a|x| = a:

    • Case 1: x=ax = a
    • Case 2: x=−ax = -a

  2. For equations like ∣x∣=−a|x| = -a (where a>0a > 0), there are no solutions since absolute values cannot equal a negative number.

  3. In more complex equations, such as ∣ax+b∣=c|ax + b| = c, the same principles apply, and it is often necessary to first isolate the absolute value expression before applying the two-case strategy.

Overall, solving absolute value equations involves recognizing the two scenarios arising from the definition of absolute value and systematically solving for the variable.

In this article
Part 1: Intro to absolute value equations and graphs Part 2: Solving absolute value equations

Part 1: Intro to absolute value equations and graphs

To solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. To graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped.

Here are the key points to focus on when studying "Intro to Absolute Value Equations and Graphs":

  1. Definition of Absolute Value:

    • Absolute value refers to the distance of a number from zero on a number line, denoted as |x|. For any real number x, |x| = x if x ≥ 0 and |x| = -x if x < 0.

  2. Absolute Value Equations:

    • General form: |expression| = k, where k ≥ 0.
    • To solve, set the expression inside the absolute value equal to k and -k:
      • expression = k
      • expression = -k

  3. Graphing Absolute Value Functions:

    • The simplest absolute value function: f(x) = |x|.
    • Shape is a “V” opening upwards, with the vertex at the origin (0,0).
    • Transformations affect the graph's position and orientation:
      • f(x) = |x - h| + k shifts the graph right/left (h) and up/down (k).
      • Scaling with a factor affects the steepness of the "V".

  4. Key Characteristics of the Graphs:

    • Vertex: The point where the graph changes direction.
    • Axis of symmetry: A vertical line that passes through the vertex.
    • Intercepts: Points where the graph crosses the x-axis and y-axis.

  5. Solving Absolute Value Inequalities:

    • Break into two cases for equations, then express in interval notation for inequalities.

  6. Applications:

    • Absolute value can model real-world situations, such as measuring quantities that cannot be negative.

  7. Practice:

    • Solve various types of absolute value equations and sketch corresponding graphs to solidify understanding.

By mastering these points, you’ll gain a solid foundation in understanding absolute value equations and their graphical representations.

Part 2: Solving absolute value equations

Understand why an absolute value equation can have from 0 to 2 solutions. Solve absolute value equations with different numbers of solutions.

When studying "Solving Absolute Value Equations," focus on these key points:

  1. Understanding Absolute Value: The absolute value of a number represents its distance from zero on the number line, disregarding the sign.

  2. Basic Format: An absolute value equation generally takes the form |A| = B, where A is an expression and B is a non-negative number.

  3. Splitting into Cases: For |A| = B, create two separate equations:

    • A = B
    • A = -B

  4. Special Cases:

    • If B < 0: The equation |A| = B has no solution since absolute values can't be negative.
    • If B = 0: The equation |A| = 0 has a single solution, A = 0.

  5. Solving Each Case: Solve each of the resulting equations and check both solutions in the original equation to ensure they are valid.

  6. Graphical Interpretation: Understand that the solution set can also be represented graphically, where the lines intersect at the points represented by the solutions.

  7. Practice Different Forms: Be familiar with equations involving expressions inside the absolute value that may require simplification or manipulation before solving.

  8. Final Checks: Always substitute your solutions back into the original equation to verify they are correct.

By mastering these points, you'll be well-equipped to tackle absolute value equations effectively.

Related topics

Solving equations with one unknown

Solutions to linear equations

Multi-step linear inequalities

Compound inequalities

Modeling with linear equations and inequalities

Part 1: Intro to absolute value equations and graphsPart 2: Solving absolute value equations