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Chapter 8: Problem 339

In the following exercises, solve each equation. $$ 2(6 x+5)-8=-22 $$

Short Answer

Expert verified
x = -2

Step by step solution

01

Distribute the 2

First, distribute the 2 in the equation by multiplying it with the terms inside the parentheses.\[2(6x + 5) - 8 = -22 \]\[12x + 10 - 8 = -22\]
02

Simplify the equation

Combine like terms on the left side of the equation.\[12x + 2 = -22\]
03

Isolate the variable term

Subtract 2 from both sides of the equation to isolate the variable term.\[12x + 2 - 2 = -22 - 2\]\[12x = -24\]
04

Solve for x

Divide both sides of the equation by 12 to solve for \(x\).\[\frac{12x}{12} = \frac{-24}{12}\]\[x = -2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It helps you to multiply a single term by each term inside a set of parentheses. For example, if you have an equation like \(2(6x + 5) - 8 = -22\), you need to distribute the 2 to both \(6x\) and 5. This results in:
\[2 * 6x + 2 * 5 - 8 = -22\text{ or } 12x + 10 - 8 = -22\]
By using the distributive property, we make it easier to combine like terms later on. Remember, always apply multiplication to every term inside the parentheses.
Combining Like Terms
Combining like terms means adding or subtracting terms that have the same variable part. This simplifies the equation and makes it easier to solve. In the equation \(12x + 10 - 8 = -22\), you can combine the constant terms (+10 and -8).
  • \(10 - 8 = 2\)
  • This changes the equation to: \(12x + 2 = -22\)
Make sure to only combine terms that have the same variable or are constants. For example, combine \(3x\) with \(5x\), or -7 with +3.
Isolating Variables
Isolating the variable means getting the variable term alone on one side of the equation, so you can easily solve for it. Starting from the equation \(12x + 2 = -22\), you need to move the constant term (+2) to the other side:
  • Subtract 2 from both sides: \(12x + 2 - 2 = -22 - 2\)
  • Simplifies to: \(12x = -24\)
Isolating variables is crucial because it helps you see what value the variable holds more clearly.
Simple Algebraic Operations
Simple algebraic operations include addition, subtraction, multiplication, and division. These operations are used to solve for the variable once it is isolated. In the equation \(12x = -24\), you need to solve for \(x\) by performing the following operation:
  • Divide both sides by 12: \(\frac{12x}{12} = \frac{-24}{12}\)
  • This simplifies to: \(x = -2\)
Make sure to perform the same operation on both sides of the equation to keep it balanced. These simple operations are the final steps to find the value of the variable.

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