Math Problem Statement

Evaluate |-4| + |4|, Find all numbers x satisfying |2x - 6| = 11, Write [2, 7) ∪ [5, 20) as a single interval, Write the following into a single interval (-∞, -10] ∪ (-∞, -8]

Solution

Let's break down and solve each of the questions from the image:

  1. Evaluate 4+4|-4| + |4|:

    • The absolute value of 4-4 is 4, and the absolute value of 4 is also 4. So: 4+4=4+4=8|-4| + |4| = 4 + 4 = 8 Answer: 8
  2. Find all numbers xx satisfying 2x6=11|2x - 6| = 11:

    • The equation 2x6=11|2x - 6| = 11 implies two cases:
      1. 2x6=112x - 6 = 11
        • Solve: 2x=17x=172=8.52x = 17 \Rightarrow x = \frac{17}{2} = 8.5
      2. 2x6=112x - 6 = -11
        • Solve: 2x=5x=52=2.52x = -5 \Rightarrow x = -\frac{5}{2} = -2.5 Answer: x=8.5x = 8.5 or x=2.5x = -2.5
  3. Write [2,7)[5,20)[2, 7) \cup [5, 20) as a single interval:

    • The union of the intervals [2,7)[2, 7) and [5,20)[5, 20) is the interval that covers both. Since these intervals overlap between 5 and 7, their union is: [2,20)[2, 20) Answer: [2,20)[2, 20)
  4. Write the following into a single interval: (,10](,8](-\infty, -10] \cup (-\infty, -8]:

    • The union of these intervals is the interval that extends from -\infty to the maximum point where both intervals overlap, which is 8-8. Hence, the single interval is: (,8](-\infty, -8] Answer: (,8](-\infty, -8]

Would you like further clarification or more details on any of these steps?

Here are 5 related questions for further practice:

  1. What is the solution to 3x5=7|3x - 5| = 7?
  2. How would you evaluate 10+3|-10| + |3|?
  3. Can you express (,5)(7,2)(-\infty, -5) \cup (-7, 2) as a single interval?
  4. Find the union of the intervals [1,4)[3,6][1, 4) \cup [3, 6].
  5. Solve x+4=9|x + 4| = 9 for xx.

Tip: Always remember that solving absolute value equations involves considering both the positive and negative cases of the expression inside the absolute value symbol!

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Solving Absolute Value Equations
Set Theory
Intervals

Formulas

Absolute value formula |a| = a if a >= 0, -a if a < 0
Absolute value equation |x - a| = b implies x - a = b or x - a = -b

Theorems

Properties of absolute value
Union of intervals

Suitable Grade Level

Grades 7-10