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Chapter 2: Problem 52

Solve each inequality and graph the solution on the number line. 2-3 x \geq 6+x

Short Answer

Expert verified
The solution to the inequality is x \leq -1. This includes -1 and all numbers less than -1.

Step by step solution

01

Simplify the Inequality

The first step involves simplifying the inequality. This is done by removing any like terms from both sides of the equation. From the inequality, 2-3x \geq 6+x, shift +x from the right side to the left side to become -3x - x and shift 2 from the left side to the right side to become 6 - 2. The simplified inequality becomes -4x \geq 4.
02

Solve for x

Now, isolate the variable x by dividing every term in the inequality by -4. However, remember that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign changes. Hence, -4x / -4 \leq 4 / -4, which simplifies further to x \leq -1.
03

Graph the Solution

To graph x \leq -1 on the number line, a closed dot is placed on -1 and a line is drawn to the left, indicating that the solution includes -1 and all numbers less than -1.

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Key Concepts

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

Inequality Simplification
Simplifying inequalities is essential in understanding and solving them. Just like equations, inequalities can involve operations like addition, subtraction, multiplication, and division on both sides. The goal here is to simplify the expression by moving terms around to have all the variable terms on one side and constants on the other.
  • Identify like terms on both sides of the inequality.
  • Use addition or subtraction to eliminate these like terms, moving them to opposite sides.
  • For example, in the inequality \(2 - 3x \geq 6 + x\), move \(+x\) to the left side making it \(-3x - x\) and 2 to the right side turning into \(6 - 2\).
  • This gives us \(-4x \geq 4\), a simpler form that is easier to solve.
Simplification transforms the inequality into a more manageable one, allowing for easier isolation of the variable.
Number Line Graphing
Number line graphing is a way to visually represent the set of values that satisfy an inequality. It provides a clear picture of which numbers are included in the solution set.
  • First, determine the main number involved in the solution. For \(x \leq -1\), this number is \(-1\).
  • Place a closed dot on \(-1\) on the number line. A closed dot means that \(-1\) is included in the solution.
  • Draw a line extending to the left of \(-1\) to cover all numbers less than \(-1\).
This graphical representation helps in quickly identifying the range of numbers that satisfy the inequality.
Variable Isolation
Variable isolation is the process of getting the variable on its own on one side of an inequality, with a straightforward number on the other side. This is crucial for determining the solution to the inequality.
  • Once the inequality is simplified, the goal is to isolate the variable. For instance, with \(-4x \geq 4\), x is multiplied by -4.
  • To isolate \(x\), divide every term by \(-4\). Remember, dividing by a negative requires flipping the inequality sign. So, \(-4x / -4 \leq 4 / -4\) leads to \(x \leq -1\).
  • This isolated form \(x \leq -1\) provides a clear set of solutions.
By isolating the variable, we identify the specific range of values that meet the inequality's criteria.

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