Problem 110 - Introductory Algebra for College Students

Question

In Exercises \(110-113\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x\) is \(-3,\) then the value of \(-3 x-9\) is \(-18\)

Step-by-Step Solution

Answer

The statement is false. The correct statement is: If \(x\) is \(-3,\) then the value of \(-3 x-9\) is \(0\).

Step 1: Substitute

Replace \(x\) in \( -3 x -9 \) with \(-3\). So, the equation becomes \( -3(-3)-9 \)

Step 2: Simplify

Calculate the expression to simplify it. \( -3(-3) = 9 \), and then subtract 9 which gives \( 9-9 = 0 \)

Step 3: Check and Correct

Compare the result with \(-18\). Since \(0 \neq -18\), the original statement is false. Therefore, the correct statement is: If \(x\) is \(-3,\) then the value of \(-3 x-9\) is \(0\).

Key Concepts

Equation Solving

Solving an equation involves finding the value of the variable that makes the equation true. In introductory algebra, we often encounter linear equations where there's usually one variable involved. By performing operations that balance both sides of the equation, we aim to isolate the variable and solve for it.

In the exercise, we're given an expression, not an equation per se, but it's common in equation solving to check statements related to expressions. This one checks if a particular value of the variable makes the statement true or false.

When solving equations or evaluating expressions, there are some key points to keep in mind:

• Always perform the same operation on both sides of an equation to maintain equality.
• Substitute the variable's value correctly when trying to evaluate or solve.
• Check your result at the end by substituting it back into the original equation or expression.

Sometimes, like in our exercise, we evaluate an expression directly with a given value to determine if it produces a true or false statement.

Substitution Method

The substitution method is a powerful tool when working with algebraic expressions and equations. Particularly useful in systems of equations, it also applies to evaluating single-variable expressions like the one in the exercise. The idea is to substitute a given value directly into the equation or expression.

Here's a simple guide to using substitution:

• Identify the variable to be replaced.
• Carefully substitute the given value for this variable in every instance within the expression.
• Simplify the expression to find the result after substitution.

In the exercise, we substituted \(x = -3\) into the expression \(-3x - 9\). This changed the expression to \(-3(-3) - 9\). Effective substitution requires accuracy; ensuring you replace every occurrence of the variable accurately impacts the correctness of the solution.

Expression Simplification

Simplifying expressions is a basic and crucial skill in algebra. It involves performing operations in a mathematical expression to reduce it to its simplest form. The goal is to make the expression more understandable and to calculate easily its value or to solve it in some problems.

There are some general tips to follow when simplifying an expression:

• Follow the order of operations, sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Combine like terms where possible, and double-check your arithmetic operations.
• Look for any unique opportunities to factor or rearrange terms that will lead to a simpler expression.

In our exercise, after the substitution of \(x = -3\), we multiply and then subtract in the expression \(-3(-3) - 9\), which simplifies to \9 - 9 = 0\. Simplification often leads to a clearer understanding of the context of an expression's relation to a problem, and can reveal whether a given expression satisfies a condition, such as our statement being false when compared to -18.