Problem 34 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ \frac{-5 y}{4}<8 $$

Step-by-Step Solution

Answer

Solution: The solution set for the inequality is all real numbers y such that $$y > \frac{-32}{5}$$.

Step 1: Write down the inequality

We have the inequality: $$ \frac{-5y}{4} < 8 $$

Step 2: Isolate the variable y

To isolate y, we'll multiply both sides of the inequality by the reciprocal of the fraction, which is -4/5. Since we're multiplying by a negative number, we must reverse the inequality symbol. We get: $$ \frac{-5y}{4} \cdot \frac{-4}{5} > 8 \cdot \frac{-4}{5} $$

Step 3: Simplify the inequality

Now, let's simplify the expression. On the left-hand side of the inequality, the coefficients -5 and -4 cancel out, giving us: $$ y > \frac{-32}{5} $$

Step 4: Write the final solution

The solution to the inequality is all real numbers y such that: $$ y > \frac{-32}{5} $$

Key Concepts

Algebraic Inequalities

Understanding algebraic inequalities is essential for solving various mathematical problems. An inequality, unlike an equation, indicates that one side is either larger or smaller than the other side, rather than equal. When solving an inequality such as \( \frac{-5y}{4} < 8 \) it's important to determine the range of values that satisfy the relationship, rather than a specific value.

When we observe the inequality sign '<', we interpret it as 'less than', which means that the quantity on the left must be smaller than that on the right. This inequality sign guides us on how to manipulate and solve the problem, keeping in mind that certain operations, such as multiplying or dividing by a negative number, will flip the sign.

Key Points to Remember

• Understand the direction of the inequality
• Operations can change the inequality's direction
• Solutions to inequalities are often ranges or sets of values

Isolating Variables

The process of isolating variables is a fundamental skill in algebra that involves rearranging an equation or inequality so the variable of interest stands alone on one side of it. In our example, we aim to isolate \(y\) in the inequality \(\frac{-5y}{4} < 8\). To achieve this, we perform operations that simplify the inequality and leave \(y\) free from constants and coefficients.

The primary goal is to get the variable by itself, providing a clear statement about its possible values. When we use reciprocal multiplication, as seen in our step-by-step solution, we're essentially undoing the initial multiplication of \(y\) by a fraction. This leads us to a simpler form where the variable is unaccompanied, giving us the inequality \(y > \frac{-32}{5}\).

Steps to Isolate a Variable

• Identify the operation involving the variable
• Perform the inverse operation on both sides
• Keep the variable positive if possible
• Ensure every step maintains the inequality's integrity

Reciprocal Multiplication

Reciprocal multiplication is a technique used to cancel out fractions when solving for a variable. It involves multiplying a number by its reciprocal so that the result is 1. A reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \) provided that neither \(a\) nor \(b\) are zero.

In the context of solving inequalities, when we have a variable like \(y\) multiplied by a fraction, \(\frac{-5}{4}\) in this case, we multiply both sides of the inequality by the reciprocal of that fraction to isolate \(y\). However, when we multiply or divide both sides of an inequality by a negative number, such as the reciprocal \(\frac{-4}{5}\), we must reverse the inequality sign to maintain the true relationship between the sides.

Important Aspects of Reciprocal Multiplication

• Reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \)
• Used to simplify fractions in equations and inequalities
• Multiplying by negative reciprocals flips the inequality