Problem 38 - Prealgebra

Question

If a number \(x\) satisfies \(x+4=-2,\) find the numerical value \(-\) of \(-3 x-2\)

Step-by-Step Solution

Answer

The numerical value is 16.

Step 1: Solve the Equation for x

The given equation is \( x + 4 = -2 \). To isolate \( x \), subtract 4 from both sides of the equation. This gives: \( x = -2 - 4 \). Simplify the right side to find \( x \).

Step 2: Simplify the Expression

Calculate \( -2 - 4 \) which equals \( -6 \). Hence, we have \( x = -6 \).

Step 3: Substitute x into New Expression

We need to find the value of \( -3x - 2 \). Substitute \( x = -6 \) into this expression: \( -3(-6) - 2 \).

Step 4: Simplify to Find the Expression Value

Calculate \( -3(-6) \) which is 18. Then calculate \( 18 - 2 \).

Step 5: Final Calculation

Subtract 2 from 18 to get 16. Therefore, \( -3x - 2 \) equals 16 when \( x = -6 \).

Key Concepts

Solving Linear Equations

Linear equations are foundational in algebra. They involve expressions set equal to a value, usually expressed in the form of \( ax + b = c \). In our original exercise, we have the linear equation \( x + 4 = -2 \). To solve for \( x \), we want \( x \) by itself on one side of the equation. We achieve this by manipulating both sides of the equation in the same way.For instance, starting with \( x + 4 = -2 \):

• Subtract 4 from both sides to move constant terms to one side. This simplifies the equation, ultimately leading to \( x = -6 \).

These steps help us isolate \( x \) because performing identical operations on both sides keeps the equation balanced. This is critical in preserving the equality and accurately solving the equation.

Substitution Method

The substitution method is a powerful tool for solving mathematical problems. It involves replacing a variable with a known value or expression, usually after solving another part of the problem. In the context of our exercise, once we determined that \( x = -6 \), we can substitute this value back into another expression to solve completely.We are tasked with finding the value of \( -3x - 2 \). Here's how substitution is applied:

• First, replace \( x \) with \(-6\) in the expression \( -3x - 2 \).
• This substitution transforms the expression to \( -3(-6) - 2 \).

By substituting, calculations become straightforward, as we replace unknown variables with numbers we have already solved for, making it easier to find the desired outcome.

Expression Simplification

Simplifying expressions is a crucial part of solving algebraic problems. This process involves reducing the expression to its simplest form. We do this by performing basic arithmetic operations such as addition, subtraction, multiplication, and division.In our problem, after substituting \( x = -6 \) into \( -3x - 2 \), we simplify as follows:

• Perform the multiplication: calculate \( -3 \times -6 \), which equals 18.
• Then proceed with the subtraction: \( 18 - 2 \).
• The final result is 16.

Simplification ensures that the expression is easy to interpret and handle, helping prevent errors in calculations. Additionally, it provides a clear path that leads to the final solution or answer.