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Chapter 1: Problem 74

Write an absolute value inequality that represents the statement. \(-11 \leq 8 p \leq 11\)

Short Answer

Expert verified
The absolute value inequality is \(|8p| \leq 11\).

Step by step solution

01

Understand the Inequality

The given compound inequality is \(-11 \leq 8p \leq 11\). This can be interpreted as a statement about the range of values that \(8p\) can take.
02

Rewrite as an Absolute Value Inequality

Recognize that the compound inequality \(-11 \leq 8p \leq 11\) implies that the distance of \(8p\) from 0 is at most 11. This is the definition of the absolute value: \(|8p| \leq 11\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

compound inequality
A compound inequality combines two inequalities that are related with AND or OR. In this exercise, it uses AND represented by \(-11 \leq 8p \leq 11\). This means both conditions must be true simultaneously. You should think of it as the intersection or overlap of the two conditions:
  • 8p must be greater than or equal to -11
  • 8p must be less than or equal to 11
Compound inequalities are quite common in algebra when describing a range of values. By understanding them, you'll see how they limit a variable to certain possible values.
absolute value
Absolute value measures the distance of a number from 0 on the number line without considering direction. It鈥檚 always positive or zero. The absolute value of x is written as \(|x|\). In our exercise, the inequality \(-11 \leq 8p \leq 11\) can be expressed using absolute value to show how far the expression can be from zero: \(|8p| \leq 11\). This means the distance of 8p from zero is less than or equal to 11.
solving inequalities
Solving inequalities involves finding all values of the variable that make the inequality true. For compound inequalities, split them into separate inequalities and solve each part. Consider this for our example:
  • \( -11 \leq 8p \)
  • \( 8p \leq 11\)
To isolate p, you divide all parts by 8:
  • \( \frac{-11}{8} \leq p \)
  • \( p \leq \frac{11}{8} \)
Thus, the solutions for p are: \( -\frac{11}{8} \leq p \leq \frac{11}{8} \).
algebra
Algebra includes operations with variables, constants, and using equations or inequalities to represent relationships or solve problems. Here, you use algebra to convert the compound inequality \(-11 \leq 8p \leq 11\) into an absolute value inequality. By recognizing it as an absolute value inequality \(|8p| \leq 11\), you simplify the problem by focusing on the boundaries and distances from zero, which makes it easier to visualize and solve.

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