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Magazine Chapter 2: Problem 42
Solve. Write the solution set using interval notation. See Examples 1 through 7. $$ \frac{3}{4}-\frac{2}{3} \geq \frac{x}{6} $$
Short Answer
Expert verified
The solution set is \((-\infty, \frac{1}{2}]\).
Step by step solution
01
Dress the inequality
To solve the inequality \( \frac{3}{4} - \frac{2}{3} \geq \frac{x}{6} \), we first need a common denominator with the terms on the left side. The denominators are 4 and 3, so our least common multiple is 12.
02
Re-write fractions with a common denominator
Convert the fractions \( \frac{3}{4} \) and \( \frac{2}{3} \) to have this common denominator:\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]
03
Solve the inequality
Now substitute these into the original inequality:\[ \frac{9}{12} - \frac{8}{12} \geq \frac{x}{6} \] Simplify the left side to get:\[ \frac{1}{12} \geq \frac{x}{6} \] Now, by finding a common denominator for both fractions (12 here), we have:\[ \frac{1}{12} \geq \frac{2x}{12} \]
04
Clear the fraction
Multiply both sides by 12 to eliminate the fraction:\[ 12 \times \frac{1}{12} \geq 12 \times \frac{2x}{12} \] This simplifies to:\[ 1 \geq 2x \]
05
Solving for x
Now solve for \( x \) by dividing both sides by 2:\[ \frac{1}{2} \geq x \] or equivalently\[ x \leq \frac{1}{2} \]
06
Write solution in interval notation
The inequality \( x \leq \frac{1}{2} \) in interval notation is:\[ (-\infty, \frac{1}{2}] \]
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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 of writing subsets of the real number line. It is a shorthand used to describe ranges of numbers, especially when dealing with inequalities or listening solution sets.
For example, if we have an inequality like \(x \leq \frac{1}{2}\), interval notation expresses this as \((-\infty, \frac{1}{2}]\). The \((-\infty, \frac{1}{2}]\) interval represents all numbers from negative infinity up to and including \(\frac{1}{2}\).
Here are some key aspects of interval notation:
For example, if we have an inequality like \(x \leq \frac{1}{2}\), interval notation expresses this as \((-\infty, \frac{1}{2}]\). The \((-\infty, \frac{1}{2}]\) interval represents all numbers from negative infinity up to and including \(\frac{1}{2}\).
Here are some key aspects of interval notation:
- Round brackets \(( \text{or} )\) mean the endpoint is not included.
- Square brackets \([ \text{or} ]\) mean the endpoint is included.
- You use a comma to separate the endpoints of the interval.
Common Denominator
Finding a common denominator is essential when you're adding or subtracting fractions with different denominators. It ensures a smooth calculation by rewriting fractions so they share the same baseline value.
To find a common denominator:
To find a common denominator:
- Identify the denominators of the fractions involved. For example, in our problem, they were 4 and 3.
- Determine the least common multiple (LCM) of these denominators. In our case, it was 12, because 12 is the smallest number that both 4 and 3 divide into without a remainder.
- Convert each fraction to an equivalent fraction with this common denominator. Multiply the numerator and denominator of each fraction by whatever factor is needed.
Fraction Simplification
Simplifying fractions is a core concept that involves reducing them to simpler forms while keeping the same value. This makes calculations easier and results clearer.
Here's how you simplify fractions:
Here's how you simplify fractions:
- Check if the numerator and the denominator of a fraction have any common factors.
- Divide both the numerator and the denominator by their greatest common factor (GCF).
- Continue this process until you can鈥檛 reduce the fraction anymore.
Solving Linear Inequalities
Solving linear inequalities involves finding the value of a variable that makes the inequality true. The steps are often similar to solving linear equations, but you must consider the properties of inequalities. Here's a general process to solve them:
- Isolate the variable term on one side of the inequality.
- If fractions are involved, find a common denominator to clear them out by multiplying both sides of the equation (remember to balance the inequality).
- Apply operations as you would in a regular equation: add, subtract, multiply, or divide.
Ensure that when multiplying or dividing by a negative number, the inequality sign is flipped. - Express the solution in interval notation if required, clearly stating the range of possible answers.
