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Chapter 9: Problem 75

Solve. $$|3 x+7| \leq 4$$

Short Answer

Expert verified
\( \ -\frac{11}{3} \leq x \leq -1\) \.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality is given as \(|3x + 7| \leq 4\). This means that the distance of \(3x + 7\) from 0 is at most 4. To solve this, consider the two possible cases for the expression inside the absolute value: \(-(3x + 7)\) and \(3x + 7\).
02

Remove the Absolute Value

The absolute value inequality \(|3x + 7| \leq 4\) can be broken down into two separate inequalities: \(-4 \leq 3x + 7 \leq 4\).
03

Solve the Combined Inequality

To solve the combined inequality \(-4 \leq 3x + 7 \leq 4\), first subtract 7 from all parts: \(-4 - 7 \leq 3x + 7 - 7 \leq 4 - 7\). This simplifies to \(-11 \leq 3x \leq -3\).
04

Isolate x

Divide each part by 3 to isolate \( x \): \(\frac{-11}{3} \leq x \leq \frac{-3}{3}\). This simplifies to \( -\frac{11}{3} \leq x \leq -1\).

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Key Concepts

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

absolute value
Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted using vertical bars. For example, \(|x|\) denotes the absolute value of \(x\). Absolute values are always non-negative. For instance: \( |5| = 5 \) \ and \( |-5| = 5 \). They measure how far away from zero a number is.
inequalities
Inequalities express a relationship between two expressions where one is not necessarily equal to the other. Signs used in inequalities include:
  • \( < \) (less than)
  • \( \leq \) (less than or equal to)
  • \( > \) (greater than)
  • \( \geq \) (greater than or equal to)
Inequalities show how quantities compare to each other.
combined inequalities
Combined inequalities involve two inequalities connected by 'and' or 'or.' They show a range of values that a variable can take. For instance, \( -4 \leq 3x + 7 \leq 4 \) is a combined inequality. It can be split into \( -4 \leq 3x + 7 \) and \( 3x + 7 \leq 4 \). Combined inequalities are useful in defining a range for the solution of an absolute value inequality.
algebraic expressions
Algebraic expressions consist of numbers, variables, and operations (like addition and multiplication). For instance, \( 3x + 7 \) is an algebraic expression. To solve inequalities involving algebraic expressions:
  • Combine like terms.
  • Isolate the variable by performing inverse operations.
This is how operations work to isolate and find the value of the variable.

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