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Magazine Chapter 2: Problem 80
Solve and graph the solution set. $$6-3|x-4|<3$$
Short Answer
Expert verified
The solution set is \(x < 3\) or \(x > 5\).
Step by step solution
01
Simplify the Inequality
Let's simplify the given inequality: \(6 - 3|x - 4| < 3\).First, subtract 6 from both sides of the inequality:\[-3|x - 4| < 3 - 6\] Simplifying further, we get:\[-3|x - 4| < -3\] Next, divide each side of the inequality by -3. Remember to reverse the inequality sign because we're dividing by a negative number:\[|x - 4| > 1\]
02
Solve the Absolute Value Inequality
The inequality \(|x - 4| > 1\) can be rewritten into a compound inequality: \[x - 4 > 1 \quad \text{or} \quad x - 4 < -1\] Let's solve each part separately.**Sub-inequality 1:**\[x - 4 > 1\] Add 4 to both sides:\[x > 5\] **Sub-inequality 2:**\[x - 4 < -1\]Add 4 to both sides:\[x < 3\] The solution set is then \(x > 5\) or \(x < 3\).
03
Graph the Solution Set
To graph the solution set, consider the results from the previous step: \(x < 3\) and \(x > 5\).1. **For \(x < 3\):** Draw a number line and shade all the numbers to the left of 3. Use an open circle at 3 because 3 is not included.2. **For \(x > 5\):** Shade all the numbers to the right of 5 on the same number line. Use an open circle at 5.The solution set is the union of these two regions on the number line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Compound Inequalities
A compound inequality is like a math statement that contains two separate inequalities joined by either "and" or "or". In the exercise given, the absolute value inequality \( |x - 4| > 1 \) converts into a compound inequality: \( x - 4 > 1 \) or \( x - 4 < -1 \). This means there are two situations we consider - one where the expression inside the absolute value is greater than 1, and another where it's less than -1.
- "And" Compound Inequality - Both parts must be true. For instance, "3 < x < 5" means \(x\) has to be more than 3 and less than 5 simultaneously.
- "Or" Compound Inequality - At least one part must be true. An example, "x < 3 or x > 5", means \(x\) can be either less than 3 or greater than 5 to satisfy the inequality.
Graphing Inequalities
Graphing inequalities involves visually displaying the solution set on a number line or coordinate plane. For our inequality, we are working with \(x < 3\) and \(x > 5\). To graph these on a number line:
- Place an open circle at the points that are not included in the solution, such as at 3 and 5 in this example. An open circle shows that the value is not part of the solution set.
- Shade the region to the left of 3, representing all values less than 3 ("x < 3"). This indicates the range of values \(x\) can be.
- Similarly, shade the region to the right of 5, representing all values greater than 5 ("x > 5").
Number Line
A number line is a straight horizontal line that has numbers placed at intervals along its length, usually represented with tick marks. It's a fundamental tool in math that can be used to graphically show the solutions to inequalities and equations.When working with inequalities:
- We use a number line to identify ranges of potential solutions. For the compound inequality \(x < 3\) or \(x > 5\), two separate regions are marked on the number line.
- Open circles at 3 and 5 demonstrate precisely where the number line represents limits beyond which the inequalities apply but do not include the points themselves.
- Shading the sections of the line indicates all the numbers that belong to the solution. The spaces between the shaded areas are not solutions.
