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Magazine Chapter 1: Problem 22
Solve the inequality. $$9 \leq 8 x-3<11$$
Short Answer
Expert verified
The solution is \([\frac{3}{2}, \frac{7}{4})\).
Step by step solution
01
Break Down the Compound Inequality
The inequality given is \(9 \leq 8x - 3 < 11\). Notice that this is a compound inequality that combines two separate inequalities: \(9 \leq 8x - 3\) and \(8x - 3 < 11\). We need to solve both inequalities separately.
02
Solve the First Inequality
For the first inequality, solve \(9 \leq 8x - 3\). Add 3 to both sides to get \(9 + 3 \leq 8x\), which simplifies to \(12 \leq 8x\). Now, divide both sides by 8 to isolate \(x\): \(\frac{12}{8} \leq x\). Simplify \(\frac{12}{8}\) to \(\frac{3}{2}\), so \(\frac{3}{2} \leq x\).
03
Solve the Second Inequality
For the second inequality, solve \(8x - 3 < 11\). Add 3 to both sides to get \(8x < 11 + 3\), resulting in \(8x < 14\). Divide both sides by 8 to isolate \(x\): \(x < \frac{14}{8}\), which simplifies to \(x < \frac{7}{4}\).
04
Combine the Results
Now, combine the solutions from Step 2 (\(\frac{3}{2} \leq x\)) and Step 3 (\(x < \frac{7}{4}\)). The overall solution is \(\frac{3}{2} \leq x < \frac{7}{4}\).
05
Express the Solution in Interval Notation
The solution \(\frac{3}{2} \leq x < \frac{7}{4}\) can be expressed in interval notation as \([\frac{3}{2}, \frac{7}{4})\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation provides a simple way to represent a set of numbers, especially when dealing with inequalities. In our original problem, the solution set for the inequality is expressed as \(\left[ \frac{3}{2}, \frac{7}{4} \right)\). Here's what each part means:
- The bracket \( [ \) indicates that \( \frac{3}{2} \) is included in the set, meaning \( x \) can be equal to \( \frac{3}{2} \).
- The parenthesis \( ) \) at \( \frac{7}{4} \) means this endpoint is not included, so \( x \) must be less than \( \frac{7}{4} \), but not equal to it.
- The entire set \( \left[ \frac{3}{2}, \frac{7}{4} \right) \) represents all numbers \( x \) that satisfy the compound inequality \( \frac{3}{2} \leq x < \frac{7}{4} \).
Solving Inequalities
Solving inequalities shares a lot of similarities with solving equations but comes with a unique rule set. The goal is to isolate the unknown variable. In the given inequality \(9 \leq 8x - 3 < 11\), we tackled it by splitting it into two simpler inequalities:
- First, we solved \(9 \leq 8x - 3\) by adding 3 to both sides and then dividing by 8, leading to \(\frac{3}{2} \leq x\).
- Next, we solved \(8x - 3 < 11\) in a similar fashion, resulting in \(x < \frac{7}{4}\).
Combining Inequalities
Combining inequalities allows us to find a range of possible values for the variable that simultaneously satisfies all given conditions. In compound inequalities like \(9 \leq 8x - 3 < 11\), both parts must be solved individually:
- The solution to the first inequality, \(\frac{3}{2} \leq x\), means that \( x \) must be at least \( \frac{3}{2} \).
- The second solution, \(x < \frac{7}{4}\), requires that \( x \) must be less than \( \frac{7}{4}\).
Mathematical Notation
Mathematical notation is a universal language used to express complex concepts succinctly. In the context of inequalities, notation like \( \leq \) and \( < \) indicates whether values are included or excluded.
- \( \leq \) means "less than or equal to", indicating inclusion of the value on the boundary.
- \( < \) signifies "less than", meaning the boundary value is not included.
