Math Problem Statement

Gambarkan grafik: a. y - 1 ≤ x ≤ y + 1 b. y = |1 - |x|| c. y ≤ |x| d. |y| < 2|x|

Solution

The question asks to "Gambarkan grafik", which translates to "Draw the graph" for each of the given inequalities or equations:

a. y1xy+1y - 1 \leq x \leq y + 1

This is an inequality describing a band of lines between two bounds:

  1. x=y1x = y - 1 (a line with slope 1 and intercept -1)
  2. x=y+1x = y + 1 (a line with slope 1 and intercept 1)

The solution region is the area between these two lines.

b. y=1xy = |1 - |x||

This is a compound absolute value function. To break it down:

  1. y=1xy = 1 - |x| gives a V-shaped graph inverted (since it opens downward) with a maximum at (0,1)(0, 1).
  2. It has an additional absolute value around the whole expression, which will reflect this V-shape upwards.

c. yxy \leq |x|

This describes the region below the graph of y=xy = |x|, which is a V-shaped graph opening upwards.

d. y<2x|y| < 2|x|

This represents two regions:

  1. y<2xy < 2|x| and y>2xy > -2|x|, which corresponds to a region between two lines: y=2xy = 2x and y=2xy = -2x.

Would you like me to sketch these graphs or provide further clarification?

Related Questions:

  1. How do you solve absolute value inequalities in general?
  2. How would the graph of y=x1y = |x - 1| differ from y=xy = |x|?
  3. What is the significance of absolute values when graphing linear inequalities?
  4. How does reflecting a graph across the x-axis or y-axis affect its shape?
  5. What are some common techniques for solving inequalities graphically?

Tip: When graphing inequalities, always shade the region representing the solution set.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Absolute Value Functions

Formulas

y - 1 ≤ x ≤ y + 1
y = |1 - |x||
y ≤ |x|
|y| < 2|x|

Theorems

Absolute Value Function Properties
Linear Inequalities

Suitable Grade Level

Grades 9-11