Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \(2 x+5>1\)

Short answer

The solution set is \((-2, \infty)\) and can be graphed by shading the region to the right of -2 on a number line.

Step-by-step solution

Isolate the variable term

Subtract 5 from both sides of the inequality to isolate the term with the variable.\[ 2x + 5 - 5 > 1 - 5 \]This simplifies to:\[ 2x > -4 \]

Solve for the variable

Divide both sides of the inequality by 2 to solve for x.\[ \frac{2x}{2} > \frac{-4}{2} \]This simplifies to:\[ x > -2 \]

Express the solution using interval notation

The inequality \( x > -2 \) can be written in interval notation as:\[ (-2, \infty) \]

Graph the solution set

On a number line, draw an open circle at -2 to show that -2 is not included in the solution set. Shade the region to the right of -2 to indicate that all numbers greater than -2 are included.

Key concepts

Isolate Variable

When solving inequalities, the first step is to isolate the variable. This means you need to get the variable on one side of the inequality by itself.
Let's look at the given inequality: \[ 2x + 5 > 1 \]
Here, the term with the variable is \( 2x \).
To isolate \( 2x \), we need to get rid of the constant term on the same side. Subtract 5 from both sides:
\[ 2x + 5 - 5 > 1 - 5 \]
This simplifies to: \[ 2x > -4 \]
Now, the variable term \( 2x \) is isolated. This step is crucial for solving the inequality.

Interval Notation

Once you have solved the inequality, you need to express the solution using interval notation.
The final solution from isolating and solving the variable was: \[ x > -2 \]
In interval notation, this is written as \[ (-2, \infty) \]
Here's how to understand interval notation:

• The parentheses \(( \) and \() \) indicate that -2 is not included (open interval).
• The symbol \( \infty \) means the solution goes on indefinitely in the positive direction.

So, the interval notation \( (-2, \infty) \) represents all numbers greater than -2 but not including -2 itself.

Graphing Inequalities

Graphing inequalities on a number line helps you visualize the solution set.
Here's how to graph the solution \( x > -2 \):

• Start by drawing a number line.
• Locate -2 on the number line.
• Draw an open circle at -2 to show that -2 is not included in the solution.
• Shade the region to the right of -2 to show that all numbers greater than -2 are included.

The graph should look like this:
\[\text{Number line: } \cdots <-3 -2(-) -1 0 1 2 3 \cdots \](Circle at -2 is open, and the line is shaded to the right of -2)
This visual representation makes it easier to understand which values are part of the solution set.

Inequality Solutions

It's important to understand what the inequality solution represents.
We solved the inequality \( 2x + 5 > 1 \) and found that \( x > -2 \).
This means any value of \( x \) greater than -2 will satisfy the original inequality.
Let's check a few values to illustrate this:

• If \( x = 0 \), then \( 2(0) + 5 = 5 > 1 \) (True)
• If \( x = -1 \), then \( 2(-1) + 5 = 3 > 1 \) (True)
• If \( x = -2.5 \), then \( 2(-2.5) + 5 = 0 \) (False)

Notice that values greater than -2 satisfy the inequality, while values less than or equal to -2 do not.
This understanding confirms that \( x > -2 \) correctly represents the solution to the inequality.