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Magazine Chapter 4: Problem 73
Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{2}{3} x+\frac{3}{2}(x-5) \leq x $$
Short Answer
Expert verified
The solution is \( x \leq \frac{45}{7} \) or \((-\infty, \frac{45}{7}]\) in interval notation.
Step by step solution
01
Simplify Expression
Start by simplifying the left-side expression of the inequality \( \frac{2}{3}x + \frac{3}{2}(x-5) \). Distribute \( \frac{3}{2} \) in \( \frac{3}{2}(x-5) \) to get \( \frac{3}{2}x - \frac{15}{2} \). Now the inequality is \( \frac{2}{3}x + \frac{3}{2}x - \frac{15}{2} \leq x \).
02
Combine Like Terms
Combine the \( x \) terms on the left side: \( \frac{2}{3}x + \frac{3}{2}x = \left(\frac{2}{3} + \frac{3}{2}\right)x \). To add these fractions, find a common denominator, which is 6: \( \frac{4}{6}x + \frac{9}{6}x = \frac{13}{6}x \). The inequality becomes \( \frac{13}{6}x - \frac{15}{2} \leq x \).
03
Move Variables to One Side
Subtract \( x \) from both sides of the inequality: \( \frac{13}{6}x - \frac{15}{2} - x \leq 0 \). This simplifies to \( \frac{13}{6}x - \frac{6}{6}x - \frac{15}{2} \leq 0 \), which simplifies further to \( \frac{7}{6}x - \frac{15}{2} \leq 0 \).
04
Solve for x
Add \( \frac{15}{2} \) to both sides: \( \frac{7}{6}x \leq \frac{15}{2} \). Now multiply both sides by the reciprocal of \( \frac{7}{6} \), which is \( \frac{6}{7} \), to isolate \( x \): \( x \leq \frac{15}{2} \cdot \frac{6}{7} \). Calculate the right side: \( \frac{15 \cdot 6}{2 \cdot 7} = \frac{90}{14} = \frac{45}{7} \), so \( x \leq \frac{45}{7} \).
05
Graph the Solution
Draw a number line with a closed circle at \( \frac{45}{7} \) to represent that \( x \) can equal \( \frac{45}{7} \), and shade to the left to represent all values less than \( \frac{45}{7} \).
06
Write in Interval Notation
The solution in interval notation is \( \left(-\infty, \frac{45}{7}\right] \) which includes all values from negative infinity up to and including \( \frac{45}{7} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
When graphing inequalities, the goal is to visualize the solution set on a number line. The inequality in this exercise, \( x \leq \frac{45}{7} \), means that \( x \) can be any value less than or equal to \( \frac{45}{7} \).
To represent this:
To represent this:
- Draw a number line with adequate spacing between each number.
- Locate \( \frac{45}{7} \) on the number line. Depending on the scale, it may be necessary to convert this fraction to a decimal.
- Place a closed circle at \( \frac{45}{7} \). A closed circle is used because the inequality sign includes "or equal to," making \( \frac{45}{7} \) part of the solution set.
- Shade the number line to the left of \( \frac{45}{7} \), indicating that all numbers less than \( \frac{45}{7} \) are solutions.
Interval Notation
Interval notation provides a concise way to describe the set of solutions for an inequality. For the inequality \( x \leq \frac{45}{7} \), the set of numbers that satisfy this condition is endless, extending from negative infinity up to \( \frac{45}{7} \). Here鈥檚 how to write that in interval notation:
- The symbol \( -\infty \) represents negative infinity, meaning there is no lower limit.
- Use a comma to separate the starting value from the stopping value in the interval.
- Since \( x \) can equal \( \frac{45}{7} \), we use a bracket \( \left]\right] \) to include this number in the interval.
Combining Like Terms
In order to solve inequalities efficiently, it's crucial to combine like terms. In this exercise, the terms involving \( x \) need to be combined on each side of the inequality. Here's how it is done:
- Identify the "like" terms. Here, they are all terms that include \( x \): \( \frac{2}{3}x \) and \( \frac{3}{2}x \).
- To combine these, first find a common denominator, which is required for adding fractions. In this problem, the common denominator for 3 and 2 is 6.
- Convert each term to this common denominator: \( \frac{2}{3} = \frac{4}{6} \) and \( \frac{3}{2} = \frac{9}{6} \).
- Add the fractions: \( \frac{4}{6}x + \frac{9}{6}x = \frac{13}{6}x \).
