Problem 20 - Prealgebra

Question

Solve each inequality. Check your answer. $$j-8 \leq-12$$

Step-by-Step Solution

Answer

The solution is \( j \leq -4 \).

Step 1: Isolate the Variable

To solve for \( j \), add 8 to both sides of the inequality to maintain the balance. This gives us:\[ j - 8 + 8 \leq -12 + 8 \]Simplifying this, we get:\[ j \leq -4 \]

Step 2: Check the Solution

To verify, substitute \( j = -4 \) into the original inequality. We have:\[ -4 - 8 \leq -12 \]Simplifying, we find:\[ -12 \leq -12 \]Since this statement is true, \( j = -4 \) satisfies the inequality. Thus, the solution is correct.

Key Concepts

Understanding and Solving Inequalities

When we talk about solving an inequality like the one in the exercise, our main goal is to find all possible values that a variable can take. This simple process involves converting the inequality into a much easier form by handling it similar to an equation. The difference here is that instead of finding just one solution, our solution pertains to a range or set of values. In our example, we started with the inequality \( j - 8 \leq -12 \).

• First, we identify what we want to find: the values of \( j \) that satisfy this condition.
• Next, we aim to simplify this inequality to get \( j \) alone on one side.
• Solving moves in simple steps - we either add, subtract, multiply, or divide both sides of the inequality by the same number, keeping in mind that multiplying or dividing by a negative reverses the inequality sign.

Often, students confuse equations with inequalities, but the big difference is in how the solution set usually isn't limited to one number but a range of numbers.

The Importance of Checking Solutions

After you find your answer, checking if it’s correct is essential. This ensures your work's accuracy and reinforces your understanding. In our example with \( j \leq -4 \), a good practice is to:

• Substitute the solution back into the original inequality.
• See if all mathematical statements still hold true.

So, by plugging \( j = -4 \) into \( j - 8 \leq -12 \), we simplified to find \( -12 \leq -12 \), which is a true statement.
In short, this check confirms that our solution \( j \leq -4 \) is indeed correct. Even though solving inequalities generally seems straightforward, checking your solution is just as important because it validates the accuracy of your work.

How to Isolate Variables in Inequalities

Isolating the variable is like peeling back layers to get to the core. Our goal here is for \( j \) to "stand alone" on one side of the inequality for easy reading and understanding. Let’s break down how this works:

• Recognize what operation is currently affecting the variable. In our example, the operation was subtraction \( j - 8 \).
• Counter the operation with its opposite: since \( j \) was subtracted by 8, we added 8 to both sides to negate the effect of subtraction.
• Make sure to perform the same operation on both sides—it’s the crucial step that ensures fairness in solving equations and inequalities.

By achieving this balance, variables are effectively isolated, making it easier to read the ensuing solution such as \( j \leq -4 \). This critical skill is applicable in many areas of math, forming a foundation for future algebraic learning.