Problem 74 - Introductory Algebra for College Students

Question

Solve and graph the solution set on a number line: \(2 x-3 \leq 5 .\) (Section \(2.7,\) Example 6 )

Step-by-Step Solution

Answer

The solution to the inequality \(2x - 3 \leq 5\) is \(x \leq 4\). On a number line, this is represented by a line extending to the left of 4, with a filled-in circle at 4.

Step 1: Rearrange the inequality to solve for \(x\)

Firstly, add 3 to each side of the inequality to remove the -3 from the left side. This will give us \(2x \leq 8\).

Step 2: Solve for \(x\)

Next, divide each side by 2 in order to isolate \(x\). This gives \(x \leq 4\).

Step 3: Graph the solution on a number line

The solution, \(x \leq 4\), means that any number smaller than or equal to 4 is a solution. On a number line, this is represented by a filled-in circle at 4 (showing that 4 is included in the solution set) then a line extending to the left (showing that any number less than 4 is part of the solution set).

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities, the main goal is to visually represent all possible solutions of an inequality. The graph gives us a clear picture of the range where solutions exist. Let's consider the inequality from our exercise: we ended up with the solution \(x \leq 4\). There are two important things to consider on the graph:

• A filled-in circle at the number 4 on the number line, indicating that 4 is included in the solution set because the inequality is \(\leq\). Always use a filled dot for inequalities that use \(\leq\) or \(\geq\) to show inclusion of the number.
  
• An arrow or line that extends to the left, representing all the numbers less than 4. This shows our solution set on the graph.
  

Learning to graph inequalities helps because it makes complex solutions more understandable and visually intuitive. Every inequality you solve can be depicted clearly through a graph. This method is particularly valuable when dealing with more intricate inequalities later on.

Algebraic Inequalities

Algebraic inequalities, much like regular equations, allow you to find a broad range of solutions rather than just a single solution. These inequalities use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to denote relationships between expressions. In solving inequalities, we employ similar principles used for equations with only a few key rules to remember. Let's reflect on our example:

• Start by simplifying the inequality. You might need to add, subtract, multiply, or divide both sides of the inequality by the same number just as in equations.
  
• When multiplying or dividing by a negative number, remember to flip the inequality sign. This is different from equations and is a crucial step.
  

In our solved inequality \(2x - 3 \leq 5\): - We added 3 to both sides to isolate the term with \(x\) which gave us \(2x \leq 8\).
- Next, we divided by 2 to fully isolate \(x\), resulting in our solution \(x \leq 4\).Understanding these steps helps in tackling any algebraic inequality by following a streamlined process that involves relaying usual equation operations.

Number Line Representation

Representing the solution to inequalities on a number line is a great visual method to showcase the range of possible values. A number line makes understanding the magnitude of our solutions more tangible. Here, we'll take our solution \(x \leq 4\) to illustrate this concept:

• Draw a horizontal line with numbers, much like a simple ruler. Make sure your number line includes 4 and other numbers around it. Remember, we are focused on illustrating where our solution resides.
  
• Place a filled circle at 4 to include it in the solution. This indicates that 4 is a part of the solutions since our inequality is \( \leq\) -- meaning it includes equality.
  
• Draw an arrow or extended line from the 4 towards the left. This shows all the numbers less than 4 are included in the set. The clear visual break from 4 to the left is vital to display the range of possible solutions logically.
  

Number line representation is fundamental in math, offering an effective way to communicate and verify your understanding of inequality solutions quickly. Once mastered, number lines make deciphering solutions much easier in a wider variety of mathematical contexts.