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Magazine Chapter 9: Problem 82
Solve. $$|x-3| \geq 2$$
Short Answer
Expert verified
x \( \leq 1 \) or x \( \geq 5 \)
Step by step solution
01
Understand the Absolute Value Inequality
The given inequality is \(|x-3| \geq 2\). This means that the expression inside the absolute value bars can either be greater than or equal to 2, or less than or equal to -2.
02
Split the Absolute Value Inequality
To solve \(|x-3| \geq 2\), split it into two separate inequalities: \(x-3 \geq 2\) and \(-(x-3) \geq 2\). Simplify the second inequality to \(x-3 \leq -2\).
03
Solve the First Inequality
Solve \(x-3 \geq 2\) by adding 3 to both sides: \(x \geq 5\).
04
Solve the Second Inequality
Solve \(x-3 \leq -2\) by adding 3 to both sides: \(x \leq 1\).
05
Combine the Solutions
Combine the two solutions to get the final answer: \(x \leq 1\) or \(x \geq 5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
solving inequalities
Solving inequalities, like the one in this exercise, involves finding all the values of the variable that make the inequality true. Unlike equations, where we find exact values, inequalities give us a range or multiple solutions.
In our given exercise, the inequality involves an absolute value, which requires us to consider both positive and negative scenarios.
In our given exercise, the inequality involves an absolute value, which requires us to consider both positive and negative scenarios.
- First, identify what the inequality is asking: \(|x-3| \geq 2\). The absolute value expression can be greater than or equal to 2 or less than or equal to -2.
- This results in two separate inequalities: \(x-3 \geq 2\) and \(-(x-3) \geq 2\).
- It's vital to solve each part individually and then combine them, as shown in steps 3 and 4.
precalculus
Precalculus serves as a bridge between algebra and calculus. It covers a range of topics including functions, equations, inequalities, and trigonometry. Understanding these concepts is crucial for success in calculus.
In our exercise, understanding absolute value functions and inequalities is critical. These are foundational topics in precalculus.
In our exercise, understanding absolute value functions and inequalities is critical. These are foundational topics in precalculus.
- An absolute value represents the distance a number is from zero on the number line, irrespective of direction (positive or negative).
- Inequalities deal with ranges of values, not just specific points.
- Mastering these concepts ensures a strong grasp on manipulating and solving different mathematical expressions, which is essential for moving forward in calculus.
step-by-step solutions
Step-by-step solutions break down complex problems into manageable parts. This makes it easier for students to follow and understand the logic behind each step.
Here鈥檚 a quick review of the steps we followed to solve our exercise:
Here鈥檚 a quick review of the steps we followed to solve our exercise:
- Step 1: Understand the inequality \( |x-3| \geq 2 \).
- Step 2: Split the inequality into two simpler inequalities: \( x-3 \geq 2 \) and \( x-3 \leq -2 \).
- Step 3: Solve each inequality separately: \( x \geq 5 \) and \( x \leq 1 \).
- Step 4: Combine the solutions to get the final answer: \( x \leq 1 \) or \( x \geq 5 \).
absolute value
Understanding absolute value is essential when solving absolute value inequalities. The absolute value of a number is its distance from zero on a number line.
Here鈥檚 what you need to know:
Here鈥檚 what you need to know:
- The absolute value of any number \(a\) is written as \(|a|\).
- This value is always non-negative because it represents distance.
- When dealing with inequalities, \(|a| \geq b\), consider both \( a \geq b \) and \( a \leq -b \).
- In our exercise, \(|x-3| \geq 2 \) means that the expression inside the absolute value bars must be either greater than or equal to 2, or less than or equal to -2.
