Problem 50 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$r+2 r<30 ; 9$$

Step-by-Step Solution

Answer

Yes, the number 9 is a solution to the inequality \(r+2r<30\).

Step 1: Simplify the Inequality

First, simplify the inequality \(r+2r<30\) to \(3r<30\). This simplification involves adding \(r\) and \(2r\) to become \(3r\).

Step 2: Substitution

Next, substitute 9 to \(r\) in the equation \(3r<30\). This results in \(3*9<30\) which simplifies further to \(27<30\).

Step 3: Verification

Finally, 27 is less than 30 and therefore, the inequality holds. So, the number 9 is a solution to the inequality \(r+2r<30\).

Key Concepts

Checking Solutions

When faced with an inequality, you may be asked to check if a specific number is a solution. To do this, you substitute the given number into the inequality and verify if the statement is true. If it holds true, then the number is indeed a solution. Here's a simple way to approach this:

• Start with the original inequality.
• Substitute the given number into the inequality where the variable is present.
• Perform any necessary operations to simplify the expression.
• If the inequality is satisfied, the number is a solution.

Using these steps, you can confirm whether your solution is correct.

Substitution Method

The substitution method is a technique often used in both equations and inequalities. It involves replacing a variable with a specific value to test if the overall mathematical statement holds true. Here's a clearer view on how substitution works in this context:

• Identify the variable in your inequality or equation.
• Replace this variable with the provided number or expression.
• Solve the inequality or equation to verify its truth.

Substitution is especially useful when checking if a number satisfies an inequality, as it allows you to "plug in" the number and quickly assess the result.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics. It requires combining like terms and reducing the expression to its simplest form. This makes it easier to solve equations or inequalities.To simplify expressions, follow these steps:

• Identify and group similar terms together, especially those with the same variable.
• Perform arithmetic operations such as addition or subtraction to combine these terms.
• Rewrite the expression in its simplest form, reducing complexity.

For example, in the inequality given above, combining terms "\(r + 2r\)" simplifies to "\(3r\)", making it easier to solve \(3r < 30\). By mastering simplification, solving mathematical problems becomes more straightforward and manageable.