Problem 35 - 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. \(y+\frac{7}{8} \leq \frac{1}{2}\)

Step-by-Step Solution

Answer

The solution to the inequality is \(y \leq -\frac{3}{8}\). The number line representation will have a filled circle at -3/8, shaded towards lower values.

Step 1: Isolate the variable

Use the addition property of inequality to get 'y' alone on one side. Subtract \(\frac{7}{8}\) from both sides of the inequality: \(y + \frac{7}{8} - \frac{7}{8} \leq \frac{1}{2} - \frac{7}{8}\)

Step 2: Simplify the inequality

Perform the subtraction operation on both sides to get: \(y \leq \frac{1}{2} - \frac{7}{8} = -\frac{3}{8}\)

Step 3: Represent solution on a number line

This is the solution in inequality form. Now, on a number line, put a closed circle on -3/8 (because the inequality is 'less than or equal to') and shade to the left (because 'y' is less than -3/8).

Key Concepts

Understanding the Addition Property of Inequality

The addition property of inequality is a valuable tool when you're solving inequalities. But what does it entail? Essentially, this property states that you can add or subtract the same value from both sides of an inequality without changing the inequality's direction. It's similar to balancing a scale.
If you add or subtract the same weight on each side, the balance remains the same. This property is crucial when you aim to isolate a variable in an inequality. For example, if you have the inequality \(y+\frac{7}{8} \leq \frac{1}{2}\), you can subtract \(\frac{7}{8}\) from both sides to keep things balanced.

• Subtract \(\frac{7}{8}\) from both sides: \(y+\frac{7}{8} - \frac{7}{8} \leq \frac{1}{2} - \frac{7}{8}\)
• This simplifies to: \(y \leq -\frac{3}{8}\)

Always remember that the inequality sign flips direction only if you multiply or divide both sides by a negative number, not when using addition or subtraction.

Graphing Inequalities

Once you have solved an inequality, it's often helpful to visualize the solution on a number line. Graphing inequalities can provide a clear representation of the solution set. Let's discuss how this works with our example inequality \(y \leq -\frac{3}{8}\).
On a number line, you'll locate \(-\frac{3}{8}\). Because the inequality is 'less than or equal to', you'll draw a closed circle at \(-\frac{3}{8}\). The closed circle indicates that \(-\frac{3}{8}\) is included in the solution set.

• Closed circle at \(-\frac{3}{8}\)
• Shade to the left, showing all numbers less than \(-\frac{3}{8}\)

By shading to the left of \(-\frac{3}{8}\), you represent all potential values for \(y\) that satisfy the inequality \(y \leq -\frac{3}{8}\). Graphing on a number line can help verify results visually and ensure that all possible solutions are clearly demonstrated.

Solving Inequalities

Solving inequalities follows a process similar to solving equations. However, there are some special considerations to take into account. The key is to isolate the variable while maintaining the inequality's balance. For the inequality \(y + \frac{7}{8} \leq \frac{1}{2}\):

• Use the addition property of inequality: subtract \(\frac{7}{8}\) from both sides.
• Perform the arithmetic: \(\frac{1}{2} - \frac{7}{8} = -\frac{3}{8}\) to isolate \(y\).

It's important to note that, unlike equations, inequalities have a few additional rules:

• If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
• Work through each step carefully, maintaining the direction of the inequality unless the above condition is met.

By following these rules and properties meticulously, you can solve inequalities effectively and with confidence, ensuring you find the correct range of possible solutions.