Math Problem Statement

Sketch the region given by the set \(\{(x, y) \mid |x| \leq 4\}\).

Solution

The problem asks you to sketch the region given by the set {(x,y)x4}\{(x, y) \mid |x| \leq 4\}.

Steps to solve:

  1. Understanding the Inequality: The inequality x4|x| \leq 4 means that the xx-coordinate of any point in the region must satisfy this condition. Since x4|x| \leq 4, this translates to 4x4-4 \leq x \leq 4.

  2. Graphing the Region:

    • The inequality 4x4-4 \leq x \leq 4 represents a vertical strip on the graph where the xx-values range between 4-4 and 44.
    • There are no restrictions on the yy-values, so yy can take any value.
    • Therefore, the region consists of all points between the vertical lines x=4x = -4 and x=4x = 4, inclusive of the boundary.
  3. Sketching:

    • Draw two vertical lines at x=4x = -4 and x=4x = 4.
    • Shade the region between these two lines, extending infinitely in the yy-direction.

This is the graphical representation of the region given by {(x,y)x4}\{(x, y) \mid |x| \leq 4\}.

Do you want any additional details or have any questions?

Related Questions:

  1. What is the graphical representation of {(x,y)y4}\{(x, y) \mid |y| \leq 4\}?
  2. How would the region change if the inequality were strict, i.e., x<4|x| < 4?
  3. What is the interpretation of x+24|x + 2| \leq 4 in terms of graphing?
  4. How would you graph the inequality {(x,y)x2+y216}\{(x, y) \mid x^2 + y^2 \leq 16\}?
  5. What does the region look like for the system of inequalities x4|x| \leq 4 and y3|y| \leq 3?

Tip: When dealing with absolute value inequalities, it is often helpful to break them down into compound inequalities for easier visualization.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Absolute Value
Graphing

Formulas

|x| \leq 4
-4 \leq x \leq 4

Theorems

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Suitable Grade Level

Grades 7-10