Inequalities with Absolute Values

This page delves into disequazioni con valore assoluto, explaining different types and solution methods for inequalities involving absolute values.

First Type of Inequalities: |x| < K

For inequalities of the form |x| < K, where K is a positive real number:

• The solution is -K < x < K

Example: Solve |x| < 11 Solution: -11 < x < 11

Second Type of Inequalities: |A(x)| > K

For inequalities of this type:

• If K > 0, the solution is A(x) < -K or A(x) > K
• If K = 0, the solution is all real numbers except where A(x) = 0
• If K < 0, the solution is all real numbers

Example: Solve |x| > 2 Solution: x < -2 or x > 2

Third Type of Inequalities: |A(x)| < K

For inequalities of this type:

• If K > 0, the solution is -K < A(x) < K
• If K = 0, the solution is A(x) = 0
• If K < 0, there is no solution

Highlight: The third type of inequality follows the same principle as equations with absolute values.

The page also covers more complex inequalities, such as those involving fractions or multiple absolute value terms.

Example: Solve |$2x + 1$ / $x + 8$| < 1 Solution: Solve the compound inequality: -1 < $2x + 1$ / $x + 8$ < 1 Solve each part separately and combine the results.

This comprehensive guide provides students with the tools to tackle a wide range of equazioni e disequazioni con valore assoluto: esercizi svolti, enhancing their understanding and problem-solving skills in this crucial area of mathematics.

Equations with Absolute Values

This page introduces the concept of absolute value and its application in equations. It covers the first two types of equations with absolute values and provides detailed examples of how to solve them.

Definition: The absolute value (or modulus) of a number is its non-negative value without regard to its sign.

Highlight: In equations with absolute values, the unknown variable is contained within the absolute value symbols.

First Type of Equations: |A(x)| = K

For equations of the form |A(x)| = K, the solution method depends on the value of K:

• If K > 0, then A(x) = ±K
• If K = 0, then A(x) = 0
• If K < 0, there is no real solution

Example: Solve |x - 1| = 6 Solution: x - 1 = 6 or x - 1 = -6 x = 7 or x = -5 Therefore, S = {-5, 7}

Second Type of Equations: |A(x)| = B(x)

For equations of this type, we set up a system of equations:

{A(x) = B(x) {A(x) = -B(x)

Example: Solve |x² + 5| = 2x Solution: Set up the system: {x² + 5 = 2x {x² + 5 = -2x Solve each equation to find S = {-5, 2}

The page concludes with a note that for x ≥ 0, |x| = x, and for x < 0, |x| = -x, which is crucial for understanding the behavior of absolute value functions.