Problem 18 - Prealgebra

Question

Solve each inequality. Check your answer. $$r-5 \leq 2$$

Step-by-Step Solution

Answer

The solution is \( r \leq 7 \).

Step 1: Understand the Inequality

We need to solve the inequality \( r - 5 \leq 2 \). This means we're trying to find all values of \( r \) that satisfy the inequality.

Step 2: Isolate the Variable

To isolate \( r \), we'll add 5 to both sides of the inequality. This gives us: \( r - 5 + 5 \leq 2 + 5 \). Simplifying this, we have \( r \leq 7 \).

Step 3: Check the Solution

Choose a value for \( r \) less than or equal to 7 and substitute it back into the original inequality to verify. Let's use \( r = 7 \): \( 7 - 5 \leq 2 \) simplifies to \( 2 \leq 2 \), which is true. Now check another value. For example, use \( r = 6 \): \( 6 - 5 \leq 2 \) simplifies to \( 1 \leq 2 \), which is also true.

Key Concepts

Prealgebra

Prealgebra is like the foundation of a house you are building in mathematics. It gives you the basic skills that are essential for understanding more complex math concepts later on.
In prealgebra, we often encounter numbers, basic equations, and inequalities. It’s all about manipulating numbers and variables to help you solve for unknowns. Here, you learn the rules of arithmetic operations like addition, subtraction, multiplication, and division.
Another important aspect of prealgebra is understanding how to deal with variables. A variable is simply a symbol, often a letter, that represents an unknown value. It's like a placeholder for a number we have yet to find. Mastering prealgebra is crucial because it sets the stage for algebra, which involves finding unknown variables using equations and inequalities. When solving problems, you may use:

• Addition and subtraction to adjust terms.
• Understanding relationships between different numbers and objects.
• Logical thinking to determine the correct process to solve a problem.

Solving Inequalities

Inequalities are expressions that involve terms being less than, greater than, less than or equal to, or greater than or equal to one another. They are similar to algebraic equations but instead of an equal sign, you use inequality symbols. Let’s break this down with the inequality from our exercise:
To solve the inequality \( r - 5 \leq 2 \), you essentially follow the same steps as solving an equation:

• First, aim to isolate the variable. This means we want \( r \) all by itself on one side of the inequality.
• To do this, we add 5 to both sides to cancel out the -5 next to \( r \).
• This results in \( r \leq 7 \).

Unlike equations, inequalities show that there are many possible solutions, not just one. Any number that is less than or equal to 7 will satisfy the inequality. This is crucial because it shows us a range of solutions, as opposed to just a single number.

Checking Solutions

Checking your solutions for inequalities is a bit like verifying the answer to make sure everything is correct. It involves substituting potential values back into the original inequality to see if they hold true.
For instance, in the inequality \( r - 5 \leq 2 \), after solving it, we found that \( r \leq 7 \). You can choose any number equal to or less than 7 and plug it back into the inequality to verify.

• Let's choose \( r = 7 \): Substitute \( 7 \) for \( r \) and see \( 7 - 5 \leq 2 \), that's \( 2 \leq 2 \), which holds true.
• Try \( r = 6 \): Substitute \( 6 \) for \( r \) and see \( 6 - 5 \leq 2 \), resulting in \( 1 \leq 2 \), also true.
• What about \( r = 8 \)? Substitute \( 8 \) for \( r \) and you get \( 8 - 5 \leq 2 \), which simplifies to \( 3 \leq 2 \), which is false, and hence not a solution.

Checking different values helps ensure the accuracy of your solution range and builds confidence in your understanding of solving inequalities.