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Magazine Chapter 4: Problem 69
Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(0 \leq \frac{4-x}{3} \leq 2\)
Short Answer
Expert verified
The solution set is \([-2, 4]\).
Step by step solution
01
Break Down the Compound Inequality
The compound inequality given is \(0 \leq \frac{4-x}{3} \leq 2\). This means we have two inequalities to solve: \(0 \leq \frac{4-x}{3}\) and \(\frac{4-x}{3} \leq 2\). We'll solve each inequality separately to find our solution set.
02
Solve the First Inequality
Start with the inequality \(0 \leq \frac{4-x}{3}\). First, eliminate the fraction by multiplying both sides by 3: \[0 \cdot 3 \leq 4-x\] or \[0 \leq 4-x\]. Then, solve for \(x\) by isolating it: \(x \leq 4\).
03
Solve the Second Inequality
Next, solve the inequality \(\frac{4-x}{3} \leq 2\). Again, eliminate the fraction by multiplying both sides by 3:\[4-x \leq 6\].Then, solve for \(x\) by isolating it: \(-x \leq 6-4\) or \(-x \leq 2\). Multiply by -1 (and remember to reverse the inequality sign): \(x \geq -2\).
04
Combine the Inequalities
Now, combine the two inequalities \(x \leq 4\) and \(x \geq -2\). The solution to the compound inequality is \(-2 \leq x \leq 4\).
05
Express in Interval Notation
The solution \(-2 \leq x \leq 4\) is expressed in interval notation as \([-2, 4]\). This interval includes all real numbers between -2 and 4, including the endpoints.
06
Graph the Solution Set
To graph \([-2, 4]\), draw a number line. Place a closed circle at -2 and another at 4 to represent the endpoints. Shade the region between the two circles to indicate all the numbers included in the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
When it comes to expressing solutions of inequalities, interval notation is a concise and clear method. This notation uses brackets to show whether endpoints are included or not.
For example, the interval \([-2, 4]\) means that all numbers starting from -2 and going up to 4 are included in the solution. Hence, the square brackets indicate that -2 and 4 are part of the set.
Key points for using interval notation are:
For example, the interval \([-2, 4]\) means that all numbers starting from -2 and going up to 4 are included in the solution. Hence, the square brackets indicate that -2 and 4 are part of the set.
Key points for using interval notation are:
- Use square brackets \([a, b]\) to include endpoints \(a\) and \(b\).
- Use parentheses \((a, b)\) for intervals where endpoints are not included.
- Combine \((,] \) or \([,)\) to mix inclusivity at one endpoint and exclusivity at the other, although they are less common in elementary level compound inequalities.
Solving Inequalities
Solving inequalities is like solving equations, but with a few critical differences, especially when dealing with compound inequalities.
Let's consider the inequalities from the example: \(0 \leq \frac{4-x}{3} \leq 2\). To solve these:
Let's consider the inequalities from the example: \(0 \leq \frac{4-x}{3} \leq 2\). To solve these:
- **Break Down the Compound:** When faced with a compound inequality like this, tackle each inequality separately. This keeps things manageable.
- **Eliminate Fractions:** If there's a fraction involved, as in both inequalities here, multiply both sides by the denominator to simplify.
- **Isolate \(x\):** Rearrange the inequality to solve for \(x\). Remember that multiplying or dividing by a negative number flips the inequality sign.
Graphing Solution Sets
Graphing solution sets helps visualize inequalities and gives us a better understanding of the range of values involved.
On a number line, \([-2, 4]\) from the previous problem can be easily graphed:
On a number line, \([-2, 4]\) from the previous problem can be easily graphed:
- **Closed Circles:** Place closed circles at the numbers -2 and 4 on the number line. These circles indicate that both -2 and 4 are included in the solution.
- **Shading the Region:** Shade the region between these two points to show that every number in this region is a solution to the inequality.
