Algebra Class 10 Test 1 Review

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.