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Magazine Chapter 1: Problem 114
Solve each three-part inequality analytically. Support your answer graphically. $$\pi \leq 5-4 x<7 \pi$$
Short Answer
Expert verified
The solution is \( \frac{5 - 7\pi}{4} < x \leq \frac{5 - \pi}{4} \).
Step by step solution
01
Break down the compound inequality
The given inequality is \[ \pi \leq 5 - 4x < 7\pi \] This can be seen as two parts: 1. \( \pi \leq 5 - 4x \)2. \( 5 - 4x < 7\pi \)
02
Solve the first part of the inequality
For \( \pi \leq 5 - 4x \), rewrite it as \[ 4x \leq 5 - \pi \]Then, solve for \( x \) by dividing both sides by 4:\[ x \geq \frac{5 - \pi}{4} \]
03
Solve the second part of the inequality
For \( 5 - 4x < 7\pi \), rearrange it as \[ -4x < 7\pi - 5 \]Multiply through by -1 and flip the inequality sign:\[ 4x > 5 - 7\pi \]Solve for \( x \):\[ x > \frac{5 - 7\pi}{4} \]
04
Combine the solutions
Combine the results from Steps 2 and 3. You have:\[ \frac{5 - 7\pi}{4} < x \leq \frac{5 - \pi}{4} \]
05
Support graphically
On a number line, mark the interval from \( \frac{5 - 7\pi}{4} \) to \( \frac{5 - \pi}{4} \). The value \( \frac{5 - 7\pi}{4} \) will have an open circle, indicating it is not included in the solution set, and \( \frac{5 - \pi}{4} \) will have a filled circle since it is included. The section of the number line between these points should be shaded to represent all values \( x \) satisfying the inequality.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Compound Inequality
A compound inequality involves two separate inequalities that must be solved together. This type of inequality often looks like a three-part expression, similar to the one in the original exercise: \( \pi \leq 5 - 4x < 7\pi \).
- This expression denotes two conditions that need to be analyzed: \( \pi \leq 5 - 4x \) and \( 5 - 4x < 7\pi \).
- Both conditions must be true simultaneously for the inequality to hold.
- The challenge is to find the values that satisfy both conditions at once.
Graphical Representation
Graphically representing inequalities helps visually confirm a solution. This is important for understanding whether a solution set has been correctly identified.
- A graph will showcase how the values relate to one another and the boundary points of the solution.
- For a three-part compound inequality like \( \pi \leq 5 - 4x < 7\pi \), visual representation can clearly indicate the valid interval for \( x \).
- Shading the appropriate portion of the number line highlights all possible solutions to the inequality.
Solving Inequalities
Solving inequalities involves a procedure somewhat similar to solving equations, but with special rules.
- The primary objective is to isolate the variable in question.
- Unlike equations, multiplying or dividing by a negative number flips the inequality sign. This is a crucial point that can easily be overlooked, leading to incorrect solutions.
- For instance, to solve the inequality \( \pi \leq 5 - 4x < 7\pi \), it was broken into two parts and solved accordingly.
Number Line
A number line can effectively illustrate the solution to an inequality. It's a simple and visual method to display an inequality's solution set.
- First, identify the solutions from the algebraic process and mark them on the number line.
- Open circles indicate numbers that aren't included in the range, while closed circles mean the number is part of the solution set.
- The area between relevant points is shaded to show all possible values of \( x \) that satisfy the inequality.
