Problem 19 - Prealgebra

Question

Solve each inequality. Check your answer. $$x-6<13$$

Step-by-Step Solution

Answer

The solution is \(x < 19\).

Step 1: Understand the Inequality

The inequality given is \(x - 6 < 13\). This means we need to find the values of \(x\) that are less than 13 when 6 is subtracted from them.

Step 2: Isolate the Variable

To find the solution for \(x\), we need to get \(x\) by itself on one side of the inequality. To do this, add 6 to both sides: \((x - 6) + 6 < 13 + 6\). This simplifies to \(x < 19\).

Step 3: Check the Solution

Pick a number less than 19, such as 18, and substitute it back into the original inequality: \(18 - 6 < 13\). Calculate: \(18 - 6 = 12\), and \(12 < 13\) is true, confirming that numbers less than 19 satisfy the inequality.

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but with a special focus on the direction of the inequality sign (like <, >, ≤, or ≥). In this case, we are solving the inequality: \( x - 6 < 13 \). The goal is to find the range of values for the variable \( x \) that make the inequality true.

To begin, isolate the variable \( x \) by performing inverse operations, meaning you do the opposite operation to both sides of the inequality. Here’s how it works:

• The inequality \( x - 6 < 13 \) suggests that 6 is being subtracted from \( x \).
• To "undo" this, add 6 to both sides of the inequality, maintaining balance.
• This gives us \((x - 6) + 6 < 13 + 6\), simplifying to the final inequality \( x < 19 \).

Now, we know that any number less than 19 will make the original inequality true. We can write the solution as \( x < 19 \), covering all possible numbers that \( x \) could be.

Checking Solutions

Once you solve an inequality, it's important to check that the solution is correct by testing it. Testing helps ensure that your solution is valid across different numbers and isn't just a lucky guess. Here’s how you can check the solution \( x < 19 \) from the original inequality \( x - 6 < 13 \):

• Choose a number less than 19, like 18, and substitute it back into the inequality.
• Replace \( x \) with 18 in the original inequality: \( 18 - 6 < 13 \).
• Calculate \( 18 - 6 \) to get 12.
• Check if 12 is indeed less than 13, which it is.

By verifying this inequality holds true, it confirms \( x < 19 \) is the correct solution, as any value chosen under this range satisfies the condition given in the original inequality.

Prealgebra Concepts

Prealgebra is all about building a strong mathematical foundation, and understanding inequalities is a vital part of that. When working with inequalities, it's crucial to understand a few basic concepts:

• **Inequality Signs**: The signs \(<, >, ≤, ≥\) are not just placeholders, but define the relationship between expressions.
• **Inverse Operations**: Just like in equations, aim to use inverse operations (adding vs. subtracting, multiplying vs. dividing) to solve for the variable.
• **Balancing the Inequality**: Whatever operation you apply to one side, you must apply to the other, maintaining the inequality's truth.
• **Checking the Result**: Always plug numbers back in to verify and confirm the solution is valid.

These concepts are not only applicable to solving simple inequalities like \( x - 6 < 13 \) but are foundational as you tackle more advanced algebra topics. Having a firm understanding of these prealgebra concepts ensures you can approach mathematical problems with confidence and accuracy.