Problem 52 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 6 y+12 \leq 5 y-1 $$

Step-by-Step Solution

Answer

Answer: The possible values of \(y\) are any value less than or equal to \(-13\).

Step 1: Subtract 5y from both sides

We will subtract \(5y\) from both sides to keep the inequality balanced: $$ 6y + 12 - 5y \leq 5y - 1 - 5y $$ This simplifies to: $$ y + 12 \leq -1 $$

Step 2: Subtract 12 from both sides

Now, we will subtract \(12\) from both sides to isolate the variable \(y\): $$ y + 12 - 12 \leq -1 - 12 $$ This simplifies to: $$ y \leq -13 $$

Step 3: Interpret the solution

The inequality \(y \leq -13\) states that any value of \(y\) less than or equal to \(-13\) will satisfy the inequality.

Key Concepts

Solving Inequalities

Inequalities are similar to equations, but instead of showing equality, they show a relationship where one side is greater or smaller than the other. In this exercise, we started with the inequality \(6y + 12 \leq 5y - 1\). The goal is to determine all possible values of \(y\) that make the inequality true. Solving inequalities often involves similar steps as solving equations, but you need to pay attention to the inequality sign. Key points to remember when working with inequalities:

• If you multiply or divide both sides by a negative number, the inequality sign flips direction.
• Keep the inequality balanced by performing the same operation on both sides.
• Check your solution by substituting back into the original inequality if necessary.

Understanding how to handle inequalities sets a strong foundation for algebra, prepping you for more complex scenarios later on.

Step-by-Step Algebra

Breaking down problems step-by-step can simplify complex algebraic expressions or inequalities. Let's reconsider the steps we followed to solve \(6y + 12 \leq 5y - 1\).

First Step: Simplify Both Sides

Initiate by subtracting \(5y\) from both sides. This action keeps the equation balanced, allowing simplification:\[6y + 12 - 5y \leq 5y - 1 - 5y\]Which then simplifies to:\[y + 12 \leq -1\]

Second Step: Further Simplification

Next, subtract \(12\) from both sides to isolate \(y\):\[y + 12 - 12 \leq -1 - 12\]This results in:\[y \leq -13\]By resolving inequalities step-by-step:

• You maintain clarity, reducing errors from frustration or oversight.
• You see how different modifications affect the expression.
• You arrive at a simplified solution systematically while understanding each logical step.

Isolation of Variables

Isolation of a variable means getting the variable on one side of the inequality or equation and everything else on the other. This is essential for problem-solving and finding the value of unknowns. In our exercise, isolating the variable \(y\) was crucial to solving \(6y + 12 \leq 5y - 1\).

The Process

• Subtract Terms: We began by subtracting \(5y\) from both sides to remove \(y\) from the right side.
• Combine Like Terms: After that, subtract \(12\) to completely isolate \(y\) on the left side.
• Result: With both steps completed, we revealed \(y \leq -13\).

Isolating a variable requires:

• Patience to work through each mathematical operation carefully.
• An understanding of balance to apply the same operations on both sides.
• Clarification through substitution to ensure your solution is indeed correct.

Mastering isolation techniques enables you to tackle more advanced algebraic problems with confidence.