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1-Variable Equations & Inequalities Unit Test

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I can't provide a test for you directly, but I can certainly help you review concepts related to one-variable equations and inequalities. Here are some key concepts and practice questions you might find useful for studying:

Concepts to Review:

  1. One-Variable Equations:

    • Standard form: Ax + B = C
    • Solving for x: Isolate the variable by performing inverse operations (addition, subtraction, multiplication, division).

  2. One-Variable Inequalities:

    • Symbols: <, >, ≤, ≥
    • Solving inequalities involves the same steps as equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
    • Graphing inequalities on a number line.

  3. Compound Inequalities:

    • Solving and graphing statements that combine two inequalities (e.g., a < x < b).

  4. Absolute Value Equations and Inequalities:

    • Equations: |x| = a leads to two possible equations: x = a or x = -a.
    • Inequalities: |x| < a and |x| > a; both have specific ways to express the solution set.

  5. Word Problems:

    • Translating real-world problems into equations or inequalities.

Practice Questions:

Equations:

  1. Solve for x: \( 2x + 3 = 11 \)
  2. Solve for x: \( 5(x - 2) = 3x + 4 \)

Inequalities:

  1. Solve for x: \( 3x - 5 > 4 \)
  2. Solve for x: \( -2(x + 3) ≤ 8 \)

Compound Inequalities:

  1. Solve for x: \( 1 < 3x + 2 < 10 \)

Absolute Value:

  1. Solve for x: \( |x - 4| = 7 \)
  2. Solve for x: \( |2x + 3| > 5 \)

Word Problem:

  1. A number decreased by 7 is at least 15. What is the number?

Sample Solutions:

  1. Solution to 1: \[ 2x + 3 = 11 \implies 2x = 8 \implies x = 4 \]

  2. Solution to 2: \[ 5(x - 2) = 3x + 4 \implies 5x - 10 = 3x + 4 \implies 2x = 14 \implies x = 7 \]

  3. Solution to 3: \[ 3x - 5 > 4 \implies 3x > 9 \implies x > 3 \]

  4. Solution to 4: \[ -2(x + 3) ≤ 8 \implies x + 3 ≥ -4 \implies x ≥ -7 \]

  5. Solution to 5: \[ 1 < 3x + 2 < 10 \implies -1 < 3x < 8 \implies -\frac{1}{3} < x < \frac{8}{3} \]

  6. Solution to 6: \[ |x - 4| = 7 \implies x - 4 = 7 \text{ or } x - 4 = -7 \implies x = 11 \text{ or } x = -3 \]

  7. Solution to 7: \[ |2x + 3| > 5 \implies 2x + 3 > 5 \text{ or } 2x + 3 < -5 \implies x > 1 \text{ or } x < -4 \]

  8. Solution to 8: Let the number be \( x \): \[ x - 7 ≥ 15 \implies x ≥ 22 \]

Feel free to work through these problems, and let me know if you have any questions or need further clarification on any topics!