Problem 45 - Prealgebra

Question

The following equations contain parentheses. Apply the distributive property to remove the parentheses, then simplify each side before using the addition property of equality. $$-(x+3)+2 x-1=6$$

Step-by-Step Solution

Answer

x = 10

Step 1: Distribute the Negative Sign

The equation given is \(-(x+3) + 2x - 1 = 6\). Apply the distributive property to remove the parentheses: \( -(x+3) = -1 \cdot (x+3) = -x - 3 \). This converts the equation to \(-x - 3 + 2x - 1 = 6\).

Step 2: Combine Like Terms on Left Side

Now, combine like terms on the left-hand side of the equation. Combine the \(-x\) and \(2x\) terms to get \(x\). Also, combine the constants \(-3\) and \(-1\) to get \(-4\). Thus the equation is simplified to: \(x - 4 = 6\).

Step 3: Use the Addition Property of Equality

To isolate \(x\), add 4 to both sides of the equation: \(x - 4 + 4 = 6 + 4\) which simplifies to \(x = 10\).

Key Concepts

Solving Equations

Solving equations is like solving a puzzle where you need to find the value of the unknown variable. In many algebraic equations, you're dealing with finding the value of "\(x\)." To solve equations effectively, you should aim to isolate the variable on one side. This means you want to have "\(x\)" by itself on either the left or the right side of the equation.

To do this, you'll often need to perform operations like addition, subtraction, multiplication, or division on both sides of the equation. These operations should always keep the equation balanced. A balanced equation ensures that any manipulation you perform retains the equation's equality, so you can solve accurately.

Imagine weighing two sides on a scale; whatever you do to one side, you must do to the other to maintain balance. As you practice more and become familiar with these operations, solving equations will become a much smoother process for you.

Like Terms

In algebra, combining like terms is a critical step in simplifying expressions and solving equations. Like terms are terms in an expression or equation that have the same variable raised to the same power. For example, "\(3x\)" and "\(5x\)" are like terms, but "\(3x\)" and "\(3y\)" are not because they have different variables.

When you're solving an equation, combining like terms helps you simplify the equation and makes it easier to solve. Look for terms with the same variables and add or subtract their coefficients.

• A simple example would be combining "\(-x\)" and "\(2x\)" to get "\(x\).""
• Similarly, grouping constants, like "\(-3\)" and "\(-1\)," helps in further simplifying an equation.

Once you've combined all the like terms, you'll likely have a much simpler equation, which is a lot easier to solve.

Addition Property of Equality

The addition property of equality is a fundamental principle in algebra that allows us to maintain an equation's balance. This property states that if you add (or subtract) the same amount to both sides of an equation, the equality still holds true.

For instance, if your equation is "\(x - 4 = 6\)," you can add "4" to both sides of the equation to isolate \(x\). By adding 4 on both sides, the left side of the equation, which was "\(x - 4\)," becomes just "\(x\)." And the right side becomes "\(10\)."

This property is particularly useful when you have a term with the variable on one side and you need to "move" a number to the other side to isolate the variable. Remember, the main goal when solving an equation is to find the value of the variable, and using the addition property effectively will help you achieve that goal by keeping the equation balanced and accurate.