Problem 42 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$3 r-15<0 ; 5$$

Step-by-Step Solution

Answer

No, 5 is not a solution to the inequality \(3r - 15 < 0\)

Step 1: Understand the inequality

The initial inequality given is \(3r - 15 < 0\), which means any valid r must satisfy this inequality for it to be a correct solution.

Step 2: Substitute the given number

Replace 'r' with 5 in the inequality: \(3(5) - 15 < 0\) which simplifies to \(15 - 15 < 0\), effectively becoming \(0 < 0\)

Step 3: Evaluate the result

Now, evaluate the result. However, the statement \(0 < 0\) is false. Therefore, 5 is not a solution to the given inequality.

Key Concepts

Checking Solutions

Checking if a number is a solution to an inequality means assessing whether the number makes the inequality true. In mathematics, inequalities are like equations, but instead of an "equals" sign, they use signs like ">", "<", ">=", or "<=" to show a relationship between two expressions. When you are given an inequality, your task is to find values that make it true.

Here’s how you do it:

• Take your given inequality.
• Substitute any proposed solution (or number) into the inequality.
• Check if the inequality holds true (i.e., if it still makes sense).

If substituting the number makes the inequality true, then the number is indeed a solution. But if it doesn’t, like when you get a false statement such as "0 < 0", the number is not a solution. This approach helps in determining the set of numbers that satisfy the inequality condition.

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In inequalities, just like in equations, algebra helps to organize and simplify problems so they can be solved efficiently. Working with algebra involves several key steps:

• Identify the structure of the inequality.
• Substitute numbers or expressions to see which make the inequality true.
• Simplify the inequality to make evaluation easier.

Algebraic manipulation, like substitution and simplification, is essential for solving inequalities. Understanding algebraic principles allows you to determine whether a certain value is a solution by precisely working through the inequality and verifying its truthfulness.

Substitution Method

The substitution method is a technique used in algebra to check potential solutions by replacing variables in an expression with specific numbers. This is a straightforward and effective method, especially in problems involving inequalities. Here is how you use the substitution method:

• Take the inequality (e.g., the one involving a variable like "r").
• Select the potential solution you want to test (e.g., a specific number).
• Replace the variable with this number.
• Simplify the inequality to check if it results in a true statement.

By using substitution, you can methodically assess whether each proposed solution holds true for the inequality. This thorough checking process is key for verifying if you have found solutions that meet the inequality's requirements or not.