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Chapter 7: Problem 91

To prepare for Chapter \(8,\) review solving inequalities \((\text {Section } 2.6)\) Solve. $$ 2(x-1) \geq 4(x+1) $$

Short Answer

Expert verified
\( x \leq - 3 \)

Step by step solution

01

Distribute

Distribute the numbers outside the parentheses on both sides of the inequality. This gives us: \[ 2 \times (x - 1) \ \rightarrow 2x - 2 \ \ \ 4 \times (x + 1) \ \rightarrow 4x + 4 \ \] Now, the inequality is: \[ 2x - 2 \geq 4x + 4 \]
02

Simplify the inequality

Next, get all terms containing \( x \) on one side and constant terms on the other side. Subtract \( 2x \) from both sides: \[ 2x - 2 - 2x \geq 4x + 4 - 2x \] This simplifies to: \[ -2 \geq 2x + 4 \]
03

Isolate the variable \( x \)

Subtract \( 4 \) from both sides to isolate the term with \( x \): \[ -2 - 4 \geq 2x \ -6 \geq 2x \]
04

Solve for \( x \)

Divide both sides by \( 2 \) to solve for \( x \): \[ \frac{-6}{2} \geq x \ -3 \geq x \] Or rather, \( x \leq -3 \)

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Key Concepts

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

Distributing Numbers
When solving inequalities, the first step often involves distributing numbers outside parentheses. This means multiplying the number outside the parenthesis by every term inside the parenthesis. For example, in the inequality \( 2(x-1) \geq 4(x+1) \), we distribute the 2 and 4 as follows:
  • Distribute 2: \( 2 \times (x - 1) = 2x - 2 \)
  • Distribute 4: \( 4 \times (x + 1) = 4x + 4 \)
As a result, the inequality becomes \( 2x - 2 \geq 4x + 4 \). This step makes it easier to work with the variables and numbers.
Isolate the Variable
The goal of solving inequalities is to find the range of values that satisfy the inequality. To achieve this, we need to isolate the variable (in this case, \( x \)) on one side of the inequality. For our example \( 2x - 2 \geq 4x + 4 \), we start by moving all terms containing \( x \) to one side and constant terms to the other:
  • Subtract \( 2x \) from both sides: \( 2x - 2 - 2x \geq 4x + 4 - 2x \rightarrow -2 \geq 2x + 4 \)
  • Subtract 4 from both sides: \( -2 - 4 \geq 2x \rightarrow -6 \geq 2x \)
Now, the variable \( x \) is isolated on the right side.
Simplify Inequality
Simplifying an inequality involves combining like terms and making it more manageable. In our example, after isolating the variable, we got \( -6 \geq 2x \). To further simplify it and solve for \( x \), we need to perform one final operation:
  • Divide both sides by 2: \( \frac{-6}{2} \geq x \rightarrow -3 \geq x \)
We can rewrite this as \( x \leq -3 \), which is the simplified form of the inequality. This step helps in understanding which values of \( x \) satisfy the inequality.
Algebraic Operations
Throughout the process of solving inequalities, algebraic operations such as addition, subtraction, multiplication, and division help manipulate the inequality to isolate the variable and simplify it. Key operations in the example:
  • Distribution: \( 2(x-1) = 2x - 2 \) and \( 4(x+1) = 4x + 4 \)
  • Subtraction: \( 2x - 2 - 2x = -2 \) and \( 4x + 4 - 2x = 2x + 4 \)
  • Further subtraction: \( -2 - 4 = -6 \)
  • Division: \( \frac{-6}{2} = -3 \)
Understanding each algebraic operation in the context of the inequality is crucial for successfully finding the solution.

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