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Magazine Chapter 2: Problem 1
For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. \(\frac{2}{5} x+\frac{1}{3} x>\frac{44}{15}\)
Short Answer
Expert verified
The solution in interval notation is \( (4, \infty) \).
Step by step solution
01
Combine Like Terms
The inequality is given as \( \frac{2}{5}x + \frac{1}{3}x > \frac{44}{15} \). Let's first combine these terms. To do this, we need a common denominator for the fractions. The least common multiple of 5 and 3 is 15. Rewriting the terms, we have \( \frac{6}{15}x + \frac{5}{15}x > \frac{44}{15} \). Combining them gives \( \frac{11}{15}x > \frac{44}{15} \).
02
Eliminate the Fraction from the Inequality
We have the inequality \( \frac{11}{15}x > \frac{44}{15} \). To eliminate the fractions, multiply both sides of the inequality by 15 (the denominator) to get rid of the fraction, yielding \( 11x > 44 \).
03
Solve for x
Now, solve the inequality \( 11x > 44 \). Divide both sides by 11 to isolate \( x \). This gives us \( x > 4 \).
04
Express the Solution in Interval Notation
The solution \( x > 4 \) means that \( x \) can be any real number greater than 4. In interval notation, this is expressed as \( (4, \infty) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way to describe a set of numbers along a number line. It's like giving a range while providing clear information about the numbers included in or excluded from that range. One key symbol used in interval notation is the parenthesis \(( )\) or brackets \([ ]\). Let's break this down:
- Parenthesis \(( )\): This symbol indicates that the endpoint is not included in the interval. For example, \((4, \infty)\) shows that all numbers greater than 4 are part of the set, but 4 itself is not included.
- Brackets \([ ]\): This indicates that the endpoint is included. For example, \([4, 10]\) means all numbers from 4 to 10, including 4 and 10.
- Infinity \((\infty)\): Represents an interval that doesn鈥檛 have an endpoint. You often use infinity with parentheses, since infinity is a concept, not a real number we can reach.
Combining Like Terms
Combining like terms is a method used to simplify expressions. It involves merging terms that have the same variable raised to the same power.
- Like terms must have the same variable part and same power. For example, \(3x\) and \(7x\) are like terms because they both have the variable \(x\), and no other power.
- To combine, simply add or subtract the coefficients of these terms. So, \(3x + 7x\) combined would become \(10x\).
Solving Inequalities
Solving inequalities is similar to solving equations but involves finding a range of values rather than a single solution. Here鈥檚 how you go about it:
- Start by isolating the variable on one side of the inequality. This might involve adding, subtracting, multiplying, or dividing both sides by the same number.
- When multiplying or dividing both sides by a negative number, remember to flip the inequality sign. For example, if \(-2x > 6\), dividing both sides by \(-2\) gives \(x < -3\).
- Keep the variable terms positive to simplify understanding.
Common Denominators
Common denominators are essential when adding or subtracting fractions, as they allow you to perform operations on the same level.
- The common denominator is basically a shared multiple of individual denominators. It helps you rewrite fractions to have the same bottom part.
- To find the common denominator, look for the least common multiple (LCM) of the denominators. For example, if the denominators are 3 and 5, the LCM is 15.
- Rewrite each term so that the fractions have this common denominator. Then, you can easily add or subtract the fractions.
