10:54
Absolute Value Equations | An Introduction | Math with Mr. J
Math with Mr. J
Overview
This video introduces the concept of absolute value equations by explaining that the absolute value of a number represents its distance from zero. It demonstrates how to solve basic absolute value equations by setting up two separate equations: one where the expression inside the absolute value equals the positive value on the other side, and another where it equals the negative value. The video walks through three examples, showing how to isolate the absolute value (though this step is simplified in the intro examples) and then solve for the variable in each of the two resulting equations. It emphasizes checking solutions by substituting them back into the original equation.
How was this?
Save this permanently with flashcards, quizzes, and AI chat
Chapters
- Absolute value measures a number's distance from zero on a number line.
- The absolute value of any number (positive or negative) is always non-negative (positive or zero).
- For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.
Understanding absolute value as distance is crucial because it explains why absolute value equations typically have two solutions.
The absolute value of 5 is 5 because it's 5 units from 0. The absolute value of -5 is also 5 because it's 5 units from 0.
- To solve an absolute value equation, first isolate the absolute value expression.
- Since the expression inside the absolute value can be positive or negative, set up two separate equations.
- Equation 1: The expression inside the absolute value equals the positive value on the other side.
- Equation 2: The expression inside the absolute value equals the negative value on the other side.
- Solve each of these two equations for the variable.
This two-equation method ensures that all possible values for the variable that satisfy the original absolute value equation are found.
For the equation |x| = 3, set up x = 3 and x = -3.
- The equation |x| = 3 has two solutions: x = 3 and x = -3.
- Both solutions can be verified by plugging them back into the original equation: |3| = 3 and |-3| = 3.
- This illustrates the core principle of absolute value equations yielding two potential answers.
This foundational example demonstrates the direct application of the absolute value definition to find solutions.
The solutions for |x| = 3 are x = 3 and x = -3.
- For the equation |m + 5| = 9, first set up two equations: m + 5 = 9 and m + 5 = -9.
- Solve the first equation (m + 5 = 9) by subtracting 5 from both sides, yielding m = 4.
- Solve the second equation (m + 5 = -9) by subtracting 5 from both sides, yielding m = -14.
- Check both solutions: |4 + 5| = |9| = 9, and |-14 + 5| = |-9| = 9.
This example shows how to solve absolute value equations when the variable is part of a larger expression, requiring additional steps to isolate the variable.
The solutions for |m + 5| = 9 are m = 4 and m = -14.
- For the equation |4x - 8| = 24, set up two equations: 4x - 8 = 24 and 4x - 8 = -24.
- Solve the first equation (4x - 8 = 24) by adding 8 to both sides, then dividing by 4, yielding x = 8.
- Solve the second equation (4x - 8 = -24) by adding 8 to both sides, then dividing by 4, yielding x = -4.
- Verify the solutions: |4(8) - 8| = |32 - 8| = |24| = 24, and |4(-4) - 8| = |-16 - 8| = |-24| = 24.
This more complex example demonstrates the order of operations needed to solve for the variable after splitting the absolute value equation.
The solutions for |4x - 8| = 24 are x = 8 and x = -4.
Key takeaways
- Absolute value represents the distance of a number from zero, which is always a positive value.
- Because distance is always positive, an absolute value equation like |expression| = positive number will typically have two solutions.
- To solve an absolute value equation, you must first isolate the absolute value expression on one side of the equation.
- After isolating the absolute value, create two separate equations: one where the expression equals the positive value, and one where it equals the negative value.
- Solve each of the two resulting equations independently to find all possible solutions.
- Always check your solutions by substituting them back into the original absolute value equation to ensure they are correct.
Key terms
Absolute ValueDistance from ZeroAbsolute Value EquationIsolateVariableSolutionInverse OperationCheck Solutions
Test your understanding
- What does the absolute value of a number represent geometrically?
- Why do absolute value equations typically result in two solutions?
- What is the first crucial step in solving any absolute value equation?
- How do you set up the two separate equations needed to solve an absolute value equation?
- Why is it important to check your solutions for an absolute value equation?