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Chapter 6: Problem 76

Solve. $$ \frac{1}{3}-x \leq \frac{2}{5} \quad $$

Short Answer

Expert verified
x \geq -\frac{1}{15}

Step by step solution

01

Isolate the variable term

To isolate the term containing the variable, subtract \frac{1}{3} from both sides of the inequality:\[\frac{1}{3} - x - \frac{1}{3} \leq \frac{2}{5} - \frac{1}{3}\].
02

Simplify the left side

After subtracting \frac{1}{3} from both sides, the inequality becomes:\[-x \leq \frac{2}{5} - \frac{1}{3}\].
03

Find a common denominator and simplify

Find the common denominator for \frac{2}{5} and \frac{1}{3}, which is 15:\[-x \leq \frac{6}{15} - \frac{5}{15}\].This simplifies to:\[-x \leq \frac{1}{15}\].
04

Solve for x

Multiply both sides by -1, remembering to reverse the inequality sign:\[x \geq -\frac{1}{15}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality properties
Understanding inequality properties is crucial when solving problems involving inequalities. Inequalities express a relationship of inequality between two expressions. Here are some key properties you should know:

  • Transitive property: If a < b and b < c, then a < c.
  • Addition property: If a < b, then a + c < b + c for any value of c.
  • Multiplication and division property: If a < b and c is a positive number, then ac < bc. However, if c is negative, then ac > bc (the inequality flips when you multiply or divide by a negative number).

When solving inequalities, these properties help guide you through each step. For example, in our problem, during the final step, we had to multiply both sides by -1 to isolate the variable x. Because we multiplied by a negative number, we reversed the inequality sign from 鈮� to 鈮� to maintain the true relationship between the expressions.
common denominators
In solving inequalities involving fractions, finding a common denominator is important. It allows you to combine or compare fractions easily.

Here's how you do it:
  • Identify the denominators of the fractions involved.
  • Find the least common denominator (LCD) of the given fractions.
  • Rewrite each fraction with the common denominator.
  • Simplify the resulting expression.

In our exercise, we had the fractions \(\frac{2}{5}\) and \(\frac{1}{3}\). The common denominator for 5 and 3 is 15. Thus, we converted both fractions:
\[\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}\text{ and } \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\]
Then, we subtracted the fractions with the common denominator, yielding the simplified expression \(\frac{1}{15}\). This simplification helped us move to the next step in isolating the variable.
isolating variables
Isolating the variable is a key step in solving both equations and inequalities. It involves manipulating the equation or inequality to get the variable on one side by itself. Here's a simple guideline on how to do it:

  • Identify the variable term: The term with your variable (e.g., -x in our example).
  • Isolate the variable term: Use inverse operations to move other terms to the opposite side. For example, in our problem, we subtracted \(\frac{1}{3}\) from both sides to isolate the variable term -x.
  • Simplify the expression: If there are fractions, find a common denominator to combine them (as we did with \(\frac{6}{15} - \frac{5}{15}\) to get \(\frac{1}{15}\)).
  • Solve for the variable: Finally, solve for the variable by performing the inverse operation, such as multiplying both sides by -1 and remembering to reverse the inequality (giving us \(x \text{ } \text{ 鈮� } -\frac{1}{15}\)).

By following these orderly steps, you'll be able to isolate the variable and solve the inequality accurately.

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