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Chapter 4: Problem 87

Solve. $$|x+6| \geq 7$$

Short Answer

Expert verified
x \geq 1 \ or \ x \leq -13

Step by step solution

01

Understand the Absolute Value Inequality

An absolute value inequality of the form \( |x + a| \geq b \) can be broken down into two inequalities: \( x + a \geq b \) or \( x + a \leq -b \).
02

Set Up the First Inequality

Start with the inequality \( x + 6 \geq 7 \). Solve for \( x \). To do this, subtract 6 from both sides: \( x + 6 - 6 \geq 7 - 6 \). Simplifying this, we get \( x \geq 1 \).
03

Set Up the Second Inequality

Start with the inequality \( x + 6 \leq -7 \). Solve for \( x \). To do this, subtract 6 from both sides: \( x + 6 - 6 \leq -7 - 6 \). Simplifying this, we get \( x \leq -13 \).
04

Combine the Solutions

Combine the results from both inequalities: \( x \geq 1 \) or \( x \leq -13 \). This represents the solution to the original absolute value inequality.

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Key Concepts

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

absolute value
The absolute value of a number is its distance from zero on the number line, regardless of direction. For any real number, the absolute value is always non-negative. Mathematically, we express the absolute value of a number x as \( |x| \). For instance:
  • \(|5| = 5\)
  • \(|-5| = 5\)
The absolute value function creates a V-shape when graphed and has a pivotal role in solving inequalities involving absolute values.
inequality
An inequality is a mathematical statement that indicates one quantity is larger or smaller than another. Unlike equations, inequalities do not assert equality. Common inequality symbols include:
  • \( > \): greater than
  • \( < \): less than
  • \( \geq \): greater than or equal to
  • \( \leq \): less than or equal to
For example, the inequality \( x > 3 \) states that x can take any value greater than 3. Solving inequalities involves finding all possible values that satisfy the condition.
solving inequalities
Solving inequalities is somewhat similar to solving equations. However, there is a critical rule: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. Let's break down the process with the inequality \( x + 6 \geq 7 \):
  • Subtract 6 from both sides: \(x + 6 - 6 \geq 7 - 6 \)
  • Simplify the inequality: \( x \geq 1 \)
This process isolates x on one side, finding the values that make the inequality true.
compound inequalities
Absolute value inequalities often lead to compound inequalities, which are two or more inequalities joined by 'and' or 'or'. The inequality \(|x+6| \geq 7\) breaks into two separate inequalities:
  • \( x + 6 \geq 7 \)
  • \( x + 6 \leq -7 \)
We solve these independently:
  • For \( x + 6 \geq 7\), subtract 6: \( x \geq 1 \)
  • For \( x + 6 \leq -7\), subtract 6: \( x \leq -13 \)
Combining these, we get the compound inequality \( x \geq 1 \) or \( x \leq -13 \). This means x falls outside the interval from -13 to 1.

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