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Magazine Chapter 8: Problem 58
Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ -4 \leq-2(x+8)<8 $$
Short Answer
Expert verified
Solution in interval notation: \((-12, -6]\).
Step by step solution
01
Understand the Compound Inequality
The inequality given is a compound inequality: \[-4 \leq -2(x + 8) < 8\]This means the inequality is composed of two parts, \(-4 \leq -2(x + 8)\)and\(-2(x + 8) < 8\). Both parts need to be solved to find the range of \(x\).
02
Solve the First Inequality
Let's start by solving the first inequality \(-4 \leq -2(x + 8)\). Divide each side by \(-2\), and remember to flip the inequality sign: \[\frac{-4}{-2} \geq x + 8\]So, \[2 \geq x + 8\]Next, subtract 8 from both sides: \[2 - 8 \geq x\]\[-6 \geq x\] or \(x \leq -6\).
03
Solve the Second Inequality
Now, solve the second inequality \(-2(x + 8) < 8\). Divide each side by -2, and flip the inequality sign: \[(x + 8) > \frac{8}{-2}\]So, \[x + 8 > -4\]Subtract 8 from both sides: \[x > -4 - 8\]\[x > -12\].
04
Combine the Solutions
Combine the solutions from both inequalities: \(x \leq -6\) and \(x > -12\).This gives us the combined inequality:\[-12 < x \leq -6\].
05
Graph the Solution Set
The solution set \(-12 < x \leq -6\) is graphically represented as a number line with a circle at -12 that is open (indicating that -12 is not included) and a circle at -6 that is closed (indicating that -6 is included). Everything in between these two points, such as x=-7, x=-8, etc., are part of the solution set.
06
Write the Solution in Interval Notation
The interval notation for \(-12 < x \leq -6\) is: \((-12, -6]\).This indicates that the set includes every number greater than -12 and up to and including -6.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
When expressing the range of solutions for inequalities, interval notation is a concise method to describe the numbers between two endpoints on a number line. Instead of listing every single number that satisfies the inequality, we use symbols to convey this efficiently.
In interval notation, brackets and parentheses play a crucial role:
In interval notation, brackets and parentheses play a crucial role:
- "(" or ")" indicates that the endpoint is not included in the solution set, often called "exclusive."
- "[" or "]" means the endpoint is included, also known as "inclusive."
Solution Set
The solution set for a compound inequality is the collection of all numbers that make the inequality true. In a sense, it's the "answer" to the inequality problem.
For the compound inequality \(-12 < x \leq -6\), the solution set includes all numbers greater than -12 but less than or equal to -6.
These values, like -11 or -7, satisfy both parts of the inequality:
For the compound inequality \(-12 < x \leq -6\), the solution set includes all numbers greater than -12 but less than or equal to -6.
These values, like -11 or -7, satisfy both parts of the inequality:
- They fit within the range defined by the inequalities \(x > -12\) and \(x \leq -6\).
- This combined solution gives us a continuous range of values, rather than individual, disconnected points.
Graphing Inequalities
Graphing inequalities offers a visual interpretation of the solution set and helps in verifying the correctness of your solution. This visualization usually employs a number line, which is straightforward and effective for simple inequalities.
For the compound inequality \(-12 < x \leq -6\):
For the compound inequality \(-12 < x \leq -6\):
- An open circle is placed at -12, indicating -12 is not included in the solution set.
- A closed circle is at -6, showing -6 is included in the solutions.
- A solid line or shading typically connects these two points on the number line, representing all the values in between that satisfy the inequality.
