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Chapter 2: Problem 13

Solve each inequality. Then graph the solution set and write it in interval notation. $$ |x-6|-7 \leq-1 $$

Short Answer

Expert verified
The solution set is \([0, 12]\) and it includes all real numbers from 0 to 12.

Step by step solution

01

Isolate the Absolute Value

Start by adding 7 to both sides of the inequality \(|x-6| - 7 \leq -1\) to isolate the absolute value expression. This gives us \(|x-6| \leq 6\).
02

Create Two Inequalities

An absolute value inequality \(|A| \leq B\) can be split into the following two compound inequalities: \(-B \leq A \leq B\). These represent the range of values inside the absolute value that satisfy the inequality. So here, we write \(-6 \leq x-6 \leq 6\).
03

Solve Each Inequality Separately

Solve the first inequality for \(x\): \(-6 \leq x-6\). Add 6 to both sides to get \(0 \leq x\). Solve the second inequality for \(x\): \(x-6 \leq 6\). Add 6 to both sides to get \(x \leq 12\).
04

Combine the Inequalities

Combine the solutions from the two inequalities to find \(0 \leq x \leq 12\), which is the range of \(x\) that satisfies the original inequality.
05

Write in Interval Notation

Write the solution \(0 \leq x \leq 12\) in interval notation as \([0, 12]\). This indicates all numbers between 0 and 12, inclusive.
06

Graph the Solution Set

Draw a number line. Plot and shade the region from 0 to 12, including both endpoints, to represent the solution set for \(x\). Use closed circles at 0 and 12 to show they are included in the solution.

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical expressions that involve the symbols ">", "<", "\geq", "\leq". They are used to compare two values and determine if one is greater, lesser, or equal to another. Unlike equations, which show equality, inequalities express a range of possible solutions. The inequality used in our exercise is the less than or equal to (\(\leq\)) type.
  • If an inequality is like "\(a > b\)", it means \(a\) is greater than \(b\).
  • If it's "\(a < b\)", then \(a\) is less than \(b\).
  • If "\(a \geq b\)", \(a\) is greater than or equal to \(b\).
  • And finally, if "\(a \leq b\)", \(a\) is less than or equal to \(b\).
Inequalities are quite useful in determining ranges of possible values instead of a specific number. This is especially helpful in real-world scenarios where conditions can vary.
Working with Interval Notation
Interval notation is a shorthand used in mathematics to describe a range of values. It comes in handy when writing out the solutions for inequalities. This notation uses brackets and parentheses to show the beginning and end of a set of numbers.
  • Square brackets (\([ ]\)) indicate that the number is included in the interval, also known as "inclusive".
  • Parentheses (\(()\)) mean the number is not included, or "exclusive".
  • For example, the interval \([0, 12]\) includes all numbers from 0 to 12, including 0 and 12.
  • Whereas \((0, 12)\) would include numbers between 0 and 12, but not 0 and 12 themselves.
Using interval notation simplifies the way we express large sets of numbers within a given interval, allowing for clear and concise communication of solutions.
Graphing Solutions of Inequalities
Graphing the solution of an inequality on a number line helps visualize the range of values that satisfy the inequality. This is a useful tool to see where the solution lies in relation to other numbers.
  • First, draw a horizontal line to represent the number line.
  • Identify and mark the important numbers from the inequality. In this case, these are 0 and 12.
  • If the inequality is "less than or equal" or "greater than or equal", use a closed circle on the number to show that it is included.
  • If it's simply "less than" or "greater than", use an open circle to show exclusion.
  • Shade the region between these points to show all numbers that satisfy the inequality.
In the exercise, the graph would have closed circles at 0 and 12, with shading in between to represent all values from 0 to 12.
Understanding Compound Inequalities
Compound inequalities combine two separate inequalities into one statement. They help define a range of values that satisfy more complex conditions. In this context, they are often found in absolute value inequalities.
  • The format "\(-B \leq A \leq B\)" indicates values for \(A\) that are between \(-B\) and \(B\), inclusive.
  • Each part of the compound inequality, \(-B \leq A\) and \(A \leq B\), must be true simultaneously.
  • In the exercise, the compound inequality is \(-6 \leq x-6 \leq 6\), which simplifies to \(0 \leq x \leq 12\).
To solve a compound inequality, handle each part separately and then combine the results for the final solution. This provides a complete range of numbers that fulfill both conditions.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set and write it in interval notation. $$ \left|\frac{x+6}{3}\right|>2 $$

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