Math Problem Statement

Solve the absolute value equations, inequalities, and implications as shown in the image.

Solution

Let's go through the questions shown in the image.

  1. First set of equations:

    • Solve the following equations: x3=x3|x - 3| = x - 3 x1=2|x - 1| = 2
  2. Second set of inequalities:

    • Solve the following inequalities: x41|x - 4| \leq 1 x+1x2\left| x + \frac{1}{x} \right| \leq 2 52xx<1\frac{5 - 2x}{x} < 1 2x+3x32|x + 3| \leq |x - 3|
  3. Third question:

    • Determine if the following implication is true: 1x3    x<1-1 \leq x \leq 3 \implies |x| < 1
  4. Fourth question:

    • Find the positive value of δ\delta so that the following implications are true: x2<δ    5x10<1|x - 2| < \delta \implies |5x - 10| < 1 x2<δ    6x18<24|x - 2| < \delta \implies |6x - 18| < 24

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Math Problem Analysis

Mathematical Concepts

Absolute Value
Inequalities
Implications

Formulas

|x - a| = b
Inequality manipulation
Implication logic

Theorems

Properties of Absolute Value
Properties of Inequalities
Implication logic

Suitable Grade Level

Grades 10-12