Solve the inequality 2𝑥 − 3𝑥 + 1 < 0 and represent the solution on a number line

Let's solve the inequality \(2x - 3x + 1 < 0\).

### Step 1: Simplify the inequality
First, combine like terms:

\[
2x - 3x = -x
\]

So the inequality simplifies to:

\[
-x + 1 < 0
\]

### Step 2: Solve for \(x\)
To isolate \(x\), subtract 1 from both sides:

\[
-x < -1
\]

Next, multiply both sides by \(-1\) to solve for \(x\). Remember, multiplying or dividing by a negative number reverses the inequality sign:

\[
x > 1
\]

### Step 3: Represent the solution on a number line
The solution \(x > 1\) can be represented on a number line as follows:

- Draw a number line.
- Mark the point \(1\) on the number line.
- Since \(x > 1\), draw an open circle at \(1\) (indicating that 1 is not included in the solution).
- Shade the line to the right of 1 to indicate all values greater than 1.

The shaded region represents all the values of \(x\) that satisfy the inequality.

### Summary:
The solution to the inequality \(2x - 3x + 1 < 0\) is \(x > 1\).

#### Would you like more details or have any questions? Here are some related questions to explore:

1. How do we solve inequalities involving absolute values?
2. What happens when we multiply or divide both sides of an inequality by a negative number?
3. How do you solve a compound inequality?
4. Can you solve and graph the inequality \(3x - 7 \geq 2x + 5\)?
5. How do you solve a quadratic inequality?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the inequality 2x - 3x + 1 < 0 step-by-step with detailed explanations and number line representation. This problem covers key algebraic concepts, including inequalities and number line representation. Suitable for students in Grades 7-9.

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