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Chapter 2: Problem 23

Solve each of the inequalities and graph the solution set on a number line. \(2 x+1<5\)

Short Answer

Expert verified
The solution is \(x < 2\); graph this by shading left from an open circle at 2.

Step by step solution

01

Isolate the Variable

First, we want to get all terms containing the variable on one side of the inequality. We start by subtracting 1 from both sides:\[2x + 1 - 1 < 5 - 1\]This simplifies to:\[2x < 4\]
02

Solve for the Variable

Next, we divide both sides of the inequality by the coefficient of the variable, which is 2. This isolates the variable:\[\frac{2x}{2} < \frac{4}{2}\]This simplifies to:\[x < 2\]
03

Graph the Solution

To graph the solution \(x < 2\) on a number line:1. Draw a number line and mark the point \(x = 2\).2. Draw an open circle at \(x = 2\) to indicate that \(2\) is not included in the solution.3. Shade the region to the left of \(2\) to represent all numbers less than 2.

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Key Concepts

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

Isolation of Variable
When solving inequalities, one of the primary goals is to isolate the variable. This allows us to clearly see the range of values that satisfy the inequality. In the provided example, the inequality is given as \(2x + 1 < 5\). Isolation of the variable is essential in figuring out what values of \(x\) make this statement true.
To do this, we start by performing inverse operations.
  • First, subtract 1 from each side of the inequality:
\[2x + 1 - 1 < 5 - 1\]This simplifies our inequality to \(2x < 4\).
  • Next, divide by 2 to completely isolate \(x\):
\[\frac{2x}{2} < \frac{4}{2}\]Now, we have isolated \(x\) with the simplified inequality \(x < 2\). Once \(x\) is isolated, we can easily interpret the solution, knowing that \(x\) represents any number less than 2.
Graphing Inequalities
After isolating the variable and determining the solution set for an inequality, it is helpful to graph the solution on a number line. This provides a visual representation, making it easier to understand the meaning of the inequality.

To graph the inequality \(x < 2\):
  • Start by drawing a horizontal line. This is your number line.
  • Identify and mark the point \(2\) on this number line.
The graph represents all values satisfying the inequality. Since \(x\) must be less than 2, and not equal to 2, you will:
  • Place an open circle at \(2\). This open circle indicates that \(x = 2\) is not part of the solution.

Shade the entire region to the left of 2 on your number line. This shading indicates all values of \(x\) that are less than 2.
Number Line
A number line serves as a simple but powerful tool to visually represent solutions of inequalities. It helps illustrate which values are included in the solution set by using specific markers like circles and lines.
Here are some important things to remember when using a number line for inequalities:
  • Identify key points: Use points that act as boundaries, like \(x = 2\) in the \(x < 2\) inequality.
  • Open circles indicate that a number is not part of the solution set, while closed circles mean the number is included.
  • Shade the region corresponding to all values that satisfy the inequality. For \(x < 2\), shade to the left of the open circle at 2.
Every time you deal with inequalities, remember that drawing a complete and clear number line can significantly enhance understanding, providing insights into the inequality's behavior.

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