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Questions and Answers
What is the solution set for the inequality |5 - x| > 7?
What value of x satisfies the equation |x - 4| = 2?
Which of the following represents the solution to |x + 6| < 4?
What is the solution set for the inequality |2x - 3| - 5 ≥ 6?
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Which value satisfies the equation |2| - 1 = 3?
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Study Notes
Absolute Value Equations
- Absolute value equations equate positive or negative outcomes of expressions.
- Example: |𝑥 − 5| = 3 gives two solutions: 𝑥 - 5 = 3 or 𝑥 - 5 = -3 leading to 𝑥 = 8 or 𝑥 = 2.
- Each equation requires separation into positive and negative scenarios.
Problem Set
- Solve these equations:
- |𝑥 - 5| = 3
- |𝑥 + 5| = 3
- |𝑥 - 4| = 2
- |𝑥 - 7| - 5 = 3 can be simplified to |𝑥 - 7| = 8, then to two solutions.
- |𝑥 + 6| = 2𝑥 - 3 involves setting expressions equal and isolating 𝑥.
- −10 − |𝑥 + 4| = −15 translates to |𝑥 + 4| = 5, generating two outcomes.
- |2| - 1 = 3 translates simply, with no dependency on 𝑥.
- Misinterpretation of |13 − |12/𝑥|| = 13 could stem from incorrect handling of absolute values.
- |5 - 2| should resolve directly.
Inequalities with Absolute Values
- Inequalities require boundary consideration for determining intervals.
- Example: |5 − 𝑥| > 7 leads to two separate scenarios producing boundaries.
- Solutions must be expressed in interval notation for clarity.
Problem Set
- Solve these inequalities:
- |5 − 𝑥| > 7
- |9 − 𝑥| ≥ 7
- |𝑥 − 3| − 2 < 7 sets up |𝑥 − 3| < 9 leading to boundaries.
- |𝑥 + 6| + 2 < 10 simplifies to |𝑥 + 6| < 8.
- |2𝑥 − 3| − 5 ≥ 6 involves isolating the absolute value.
- |−4𝑥 − 16| − 60 ≤ 4 may need setup in positive/negative scenarios.
- 5|𝑥 − 3| < 0.004 implies |𝑥 - 3| must be less than a small threshold.
- 3 − |2| < 2 reflects direct numerical comparisons involving constants.
Notation
- Ensure all final answers are presented in interval notation for inequalities.
- Absolute value gives rise to ranges rather than single solutions in inequality contexts.
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Description
Prepare for your Algebra Class 10 Test 1 with this comprehensive review quiz. This quiz covers solving absolute value equations and inequalities, ensuring you understand the concepts required for success. Test your skills in interval notation as well as solving equations step-by-step.
