Problem 36 - Introductory Algebra for College Students

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. \(y+\frac{1}{3} \leq \frac{3}{4}\)

Step-by-Step Solution

Answer

The solution to the given inequality \(y+\frac{1}{3} \leq \frac{3}{4}\) is \(y \leq \frac{5}{12}\)

Step 1: Subtract \(\frac{1}{3}\) from both sides of the inequality

The operation should give us \(y \leq \frac{3}{4} - \frac{1}{3}\).

Step 2: Simplify the right side

To simplify the right side of the inequality, calculate the difference between \(\frac{3}{4}\) and \(\frac{1}{3}\). This gives us \(y \leq \frac{5}{12}\).

Step 3: Graph the solution set on a number line

Plot \(\frac{5}{12}\) on a number line. Because the original inequality includes equals, the point \(\frac{5}{12}\) itself is included in the solution, and the circle denoting this point is filled in.

Key Concepts

Addition Property of Inequality

Inequality problems often require us to isolate a variable on one side, much like we do in equations. The addition property of inequality is a fundamental concept for solving inequalities. It states that if you add or subtract the same number from both sides of an inequality, the relationship between the sides does not change. This means if we have an inequality like \(a < b\), and we add or subtract a number \(c\) from both sides, we'd get \(a \pm c < b \pm c\).

For the inequality \(y + \frac{1}{3} \leq \frac{3}{4}\), we want to isolate \(y\). By subtracting \(\frac{1}{3}\) from both sides, following this addition property, we are left with \(y \leq \frac{3}{4} - \frac{1}{3}\), moving us one step closer to finding the range of possible values for \(y\).

Number Line Graphing

When working with inequalities, number line graphing is a visual aid to represent the set of numbers that are solutions to the inequality.

A number line is essentially a line with marks at regular intervals, each representing a number. To graph the inequality solution \(y \leq \frac{5}{12}\), first locate the point corresponding to \(\frac{5}{12}\) on the number line. Because \(y\) is less than or equal to \(\frac{5}{12}\), every number to the left of this point is also a solution. To indicate \(y\) includes \(\frac{5}{12}\), we fill in the circle at that point. Then, draw a line or arrow extending to the left to show the continuous set of numbers that satisfy the inequality. Number line graphing provides an intuitive and efficient way to communicate and understand the solutions to inequalities.

Simplifying Fractions

In solving inequalities, you may come across the need for simplifying fractions, as these can make an inequality more straightforward to understand and solve. Simplifying a fraction means to reduce it to its lowest terms, where the numerator and the denominator have no common factors other than one.

For instance, to simplify the fraction \(\frac{3}{4} - \frac{1}{3}\), we need a common denominator. To find it, look for the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 3 is 12. Rewrite each fraction with the common denominator so that \(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\) and \(\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}\). Now, subtract the two to get \(\frac{9}{12} - \frac{4}{12} = \frac{5}{12}\), which is already in simplest form. Remember, simplifying fractions is crucial for accurately solving and graphing inequalities.