Problem 13 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ 5 y+7 \leq 22 $$

Step-by-Step Solution

Answer

Question: Solve the inequality 5y + 7 ≤ 22 and provide the solution in interval notation. Solution: The solution to this inequality is y ≤ 3. In interval notation, this is written as (-∞, 3].

Step 1: Subtract 7 from both sides

We start by subtracting 7 from both sides of the inequality. This will eliminate the constant term on the left side: $$ 5y + 7 - 7 \leq 22 - 7 $$ Simplifying, we get: $$ 5y \leq 15 $$

Step 2: Divide both sides by 5

Now, we'll divide both sides of the inequality by 5 to isolate the variable \(y\): $$ \frac{5y}{5} \leq \frac{15}{5} $$ Simplifying, we get the solution to the inequality: $$ y \leq 3 $$

Key Concepts

Inequality Operations

Understanding inequality operations is foundational in solving inequality equations. Inequalities express a relationship where one side is not necessarily equal to the other, often signified by symbols like \( \leq, \lt, \geq, \gt \). When solving inequalities, the operations you perform must keep the inequality true. For instance, consistently applying the same operation to both sides is key.

• **Addition/Subtraction**: You can add or subtract the same number from both sides without changing the inequality's direction.
• **Multiplication/Division**: If you multiply or divide both sides by a positive number, the direction of the inequality stays the same. However, if you multiply or divide by a negative number, the inequality must flip its direction.

These principles help maintain the balance and truth of the inequality as you work through various algebraic manipulations.

Isolation of Variable

Isolation of a variable is a critical step in solving inequalities. The goal here is to get the variable by itself on one side of the inequality, which allows you to easily see its possible values. Here's the approach for our exercise:

In the inequality \(5y + 7 \leq 22\), the variable \(y\) is entangled with other terms. The first move is to eliminate the constant term on the same side as \(y\) by using subtraction or addition. In this case, we subtract 7 from both sides:

\[ 5y + 7 - 7 \leq 22 - 7 \]

This simplification results in \(5y \leq 15\). Now, the variable \(y\) is multiplied by 5, so the final step is to divide both sides by 5 to isolate the variable:

\[ \frac{5y}{5} \leq \frac{15}{5} \]

The result of these operations gives us the inequality solution \(y \leq 3\). Isolating variables allows you to see the conditions that \(y\) must satisfy.

Simplification Steps

Simplification steps help make an inequality easier to solve and involve reducing the problem into its simplest form. Breaking down the inequality into manageable components clarifies the process and solution path.

Consider the inequality \(5y + 7 \leq 22\). Here are the steps to simplify:

• **Step 1**: Subtract 7 from both sides to remove the constant term on the left. This leads us from \(5y + 7 \leq 22\) to a simpler form, \(5y \leq 15\).
• **Step 2**: Divide every term by 5 to isolate \(y\), transitioning to our final simplified form, \(y \leq 3\).

These simplification steps are crucial as they progressively bring the inequality to a point where the solution is obvious and straightforward. This methodical approach ensures clarity at each stage and accuracy in your final answer.