Problem 110 - Introductory and Intermediate Algebra for College Students 4th

Question

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. A more correct statement would be 'If \(x\) is -3, then the value of \(-3x-9\) is 0.'

Step 1: Substitute the given value

Substitute \(x = -3\) in the given expression, we have \( -3(-3) -9\)

Step 2: Simplify the expression

After performing the multiplication and the subtraction, we get \(9 - 9 = 0\)

Step 3: Verify the statement

The value obtained after simplification is 0, but in the provided statement the value of the expression is given as -18, so the statement is false

Key Concepts

Expressions

In algebra, an expression is a combination of numbers, variables, and operators like addition, subtraction, multiplication, and division. The expression in this exercise is \(-3x - 9\). Algebraic expressions are used to represent mathematical situations. They are different from equations, as they do not contain an equal sign.

To work with expressions, it is important to understand how each part interacts. Here, \(-3x\) means \(-3\) multiplied by the variable \(x\). The \(-9\) is a constant that is subtracted from the result of \(-3x\).

• Expressions can be simplified by combining like terms or using arithmetic operations.
• Recognizing expressions helps in understanding and solving algebra problems. Core concepts like variables and coefficients play a key role.

Learning to analyze expressions allows you to substitute values and simplify complex problems efficiently.

Substitution

Substitution is an essential concept in algebra which involves replacing a variable with its given value. In this exercise, you substitute \(x = -3\) into the expression \(-3x - 9\).

By substituting, you turn the expression into a simple arithmetic problem. For example, replacing \(x\) with \(-3\) gives us: \(-3(-3) - 9\).

• Substitution is useful for evaluating expressions and solving equations.
• It helps in determining the value of an expression or verifying if a solution is accurate.

After substitution, it's crucial to carefully follow the order of operations to get the correct outcome. The multiplication needs to be done before the subtraction. This ensures that the expression is calculated accurately.

Simplification

Simplification refers to the process of performing all arithmetic operations to reduce an expression to its simplest form. After substituting the value of \(x\) in \(-3x - 9\), we obtained \(-3(-3) - 9\).

Simplification makes expressions easier to understand and work with. Here, \(-3\) times \(-3\) equals \(9\), and then subtracting \(9\) gives zero as the simplified form: \(9 - 9 = 0\).

• Knowing how to simplify can transform complex expressions into manageable calculations.
• It's crucial for solving equations and validating algebraic statements.

Always complete multiplication and division first, followed by addition and subtraction. This adherence to order of operations ensures accuracy in simplification. By mastering simplification, you enhance your problem-solving abilities significantly.