Kategoriak: All - inequalities - expression - solutions - error

arabera David Kedrowski 14 years ago

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MAT.105X 9.2

Absolute value inequalities are explored through various scenarios, each yielding distinct solution sets based on the nature of the inequality. When the absolute value of an expression is compared to a non-positive number, it often results in either no solution or a specific solution.

MAT.105X 9.2

MAT.105X 9.2 Solving Absolute Value Inequalities

Solve Special Cases of Absolute Value Inequalities

>= 0

Given the inequality |P| >= 0 where P is an expression, any real number is a solution. That is, the solution set consists of all real numbers.

Note that |P| > 0 will have one value that does not work as a solution, namely the value that makes P = 0. All other real numbers will work.

> or >= and k < 0

Given the inequality |P| >= 0 where P is an expression and k is any negative real number, the solution set consists of all real numbers. (>= can be replaced with >.)

<= 0

Given the inequality |P| <= 0 where P is an expression, the solution will consist only of the solution to P = 0.

Note that |P| < 0 has no solution.

< or <= and k < 0

Given the inequality |P| <= k where P is an expression and k is any negative real number, there is no solution. (<= can be replaced with <.)

Solve an Applied Problem Using an Absolute Value Inequality

Error

Measurements usually have a certain amount of uncertainty, or possible error. If a given measurement is supposed to be m but can be k less than m or k greater than m, we often say the measurement is m +- k.

If x is the actual measurement of a specific instance, then we can write

m - k <= x <= m + k.

Which can be written as the absolute value inequality

|x - m| <= k.

Solve Absolute Value Inequalities Involving > or >=

Let P be an expression and let k be a positive real number. To solve |P| >= k, solve the compound inequality P >= k or P <= -k. (> may be substituted for >=.)

Solve Absolute Value Inequalities Containing < or <=

Solving

Let P be an expression and let k be a positive real number. To solve |P| <= k, solve the three-part inequality -k <= P <= k. (< may be substituted for <=.)

Definition

An absolute inequality is a mathematical statement that includs an inequality (<, <=, >, or >=) and the absolute value of an expression involving a variable.