Problem 43 - Prealgebra

Question

Solve each inequality. Check your solution. $$g-1 \frac{2}{3}>2 \frac{1}{6}$$

Step-by-Step Solution

Answer

The solution is \( g > \frac{23}{6} \).

Step 1: Convert Mixed Numbers to Improper Fractions

The given inequality is \( g - 1 \frac{2}{3} > 2 \frac{1}{6} \). Start by converting the mixed numbers to improper fractions. The number \( 1 \frac{2}{3} \) is equivalent to \( \frac{5}{3} \) and \( 2 \frac{1}{6} \) is equivalent to \( \frac{13}{6} \).

Step 2: Rewrite the Inequality

Substitute the improper fractions into the inequality: \( g - \frac{5}{3} > \frac{13}{6} \).

Step 3: Eliminate the Fraction from the Subtraction

Add \( \frac{5}{3} \) to both sides of the inequality to isolate \( g \). We need a common denominator to add fractions, which is 6. \( \frac{5}{3} \) is equivalent to \( \frac{10}{6} \).

Step 4: Solve for g

Add \( \frac{10}{6} \) to both sides: \[ g > \frac{13}{6} + \frac{10}{6} \]Combine the fractions on the right side:\[ g > \frac{23}{6} \].

Step 5: Conclusion

The solution is \( g > \frac{23}{6} \). This means any value of \( g \) greater than \( \frac{23}{6} \) will satisfy the inequality.

Step 6: Verify the Solution

To check the solution, choose a value greater than \( \frac{23}{6} \) for \( g \), such as 5. Substitute \( g = 5 \) into the original inequality:\[ 5 - \frac{5}{3} > \frac{13}{6} \]This simplifies to:\[ \frac{15}{3} - \frac{5}{3} = \frac{10}{3} = \frac{20}{6} \]Since \( \frac{20}{6} > \frac{13}{6} \), the solution is verified.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used in everyday life because they make it easier to express fractions greater than one in a clear and understandable way. For example, when baking, one might use "1 1/3 cups of flour" instead of its improper fraction equivalent.

• The whole number and fraction are separated by a space.
• Example: 2 1/4 means 2 whole units and an additional quarter of a unit.

To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator of the fraction to this product.
• Place this sum over the original denominator.

For instance, converting 1 \(\frac{2}{3}\) to an improper fraction involves multiplying 1 by 3 and adding 2, resulting in \(\frac{5}{3}\).

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This type of fraction represents a value greater than or equal to one whole unit, which can sometimes appear less intuitive than mixed numbers.

• Example: \(\frac{7}{4}\) represents more than one whole, as the numerator 7 is greater than the denominator 4.
• To visualize, think of having 7 quarters out of groups of 4, allowing for more than a full group.

Converting improper fractions to mixed numbers can make them easier to understand:

• Divide the numerator by the denominator.
• The quotient becomes the whole number, and the remainder is the new numerator.
• Place the remainder over the original denominator.

Using the example \(\frac{7}{4}\), dividing 7 by 4 gives a quotient of 1 with a remainder of 3, resulting in the mixed number 1 \(\frac{3}{4}\).

Solving Inequalities

Inequalities are mathematical statements that compare expressions, using symbols like "\(>\)" (greater than), "\(<\)" (less than), "\(\geq\)" (greater than or equal to), and "\(\leq\)" (less than or equal to). Solving inequalities involves finding the set of values that make the inequality true.
Often, solving inequalities requires converting to improper fractions to make calculations more straightforward, especially when dealing with mixed numbers.
Here's a step-by-step strategy:

• Isolate the variable: Rearrange the equation so that the variable is on one side. This often involves adding or subtracting terms on both sides.
• Use common denominators: When adding or subtracting fractions, use equivalent fractions to match denominators for straightforward calculation.
• Check your solution: Choose a test value for the variable that satisfies the resulting inequality to ensure accuracy.

For example, solving the inequality \(g - \frac{5}{3} > \frac{13}{6}\), adding \(\frac{5}{3}\) to both sides results in \(g > \frac{23}{6}\). Checking with a test value, such as \(g = 4\), can confirm the solution's validity.