Solving Two-Step Inequalities

When it comes to solving two-step inequalities, the process is similar to solving two-step equations. However, one crucial aspect to remember is how the direction of the inequality sign reacts if you multiply or divide both sides by a negative number. In this article, we will dive into the techniques for solving two-step inequalities and how to graph their solutions.

Understanding Two-Step Inequalities

A two-step inequality is an inequality that requires two operations to isolate the variable. The general form of a two-step inequality may look like this:

\[ ax + b < c \]

Where:

  • \( a \) is a coefficient,
  • \( x \) is the variable, and
  • \( b \) and \( c \) are constants.

The goal is to isolate \( x \) on one side of the inequality while preserving the inequality's direction.

Key Inequality Symbols

Before we dive into the methods, let’s quickly review the inequality symbols you'll encounter:

  • \( < \) means "less than"
  • \( > \) means "greater than"
  • \( \leq \) means "less than or equal to"
  • \( \geq \) means "greater than or equal to"

These symbols change how we interpret the solutions of our inequalities.

Steps to Solve Two-Step Inequalities

Let's break down the steps to solve a two-step inequality clearly and effectively.

Step 1: Identify and Isolate the Variable

Start with the two-step inequality. For example:

\[ 2x + 3 < 11 \]

Your first task is to isolate the term containing \( x \). To do so, subtract the constant from both sides of the inequality:

\[ 2x + 3 - 3 < 11 - 3 \]

This simplifies to:

\[ 2x < 8 \]

Step 2: Solve for the Variable

Now, you’ll want to divide both sides by the coefficient of \( x \):

\[ \frac{2x}{2} < \frac{8}{2} \]

This gives you:

\[ x < 4 \]

Step 3: Write the Solution

The solution to the inequality \( 2x + 3 < 11 \) is \( x < 4 \). You can express it in interval notation as:

\[ (-\infty, 4) \]

This indicates that \( x \) can take any value less than 4.

Example 1: Solving with \( > \)

Let’s look at another example:

\[ 3x - 5 > 7 \]

  1. Isolate the variable: Add 5 to both sides. \[ 3x - 5 + 5 > 7 + 5 \] Simplifying gives: \[ 3x > 12 \]

  2. Solve for the variable: Divide both sides by 3. \[ \frac{3x}{3} > \frac{12}{3} \] Which simplifies to: \[ x > 4 \]

  3. Write the solution: In interval notation, the solution is: \[ (4, \infty) \]

Example 2: Dealing with Negative Coefficients

Now, let’s check what happens when we have a negative coefficient:

\[ -2x + 6 \leq 10 \]

  1. Isolate the variable: Subtract 6 from both sides. \[ -2x + 6 - 6 \leq 10 - 6 \] Which simplifies to: \[ -2x \leq 4 \]

  2. Solve for the variable: Divide by -2. Remember, when you divide or multiply by a negative number, you must flip the inequality sign. \[ \frac{-2x}{-2} \geq \frac{4}{-2} \] This gives: \[ x \geq -2 \]

  3. Write the solution: In interval notation, you express the result as: \[ [-2, \infty) \]

Step 4: Graphing the Solution

Once you have the solution, you may choose to graph the inequality on a number line.

  • For \( x < 4 \), you would draw an open circle at 4 (since 4 is not included) and shade to the left.
  • For \( x > 4 \), you would draw an open circle at 4 and shade to the right.
  • If your solution includes equal to, such as \( x \geq -2 \), you would draw a closed circle at -2 (indicating that -2 is part of the solution) and shade to the right.

Graphing visually represents your solution, making it easier to see what values satisfy the given inequality.

Common Mistakes to Avoid

While solving two-step inequalities, here are a few common pitfalls to watch for:

  1. Flipping the Inequality: Remember to flip the sign only when multiplying or dividing by a negative number.

  2. Skipping Steps: It’s essential to maintain clarity by writing out each step rather than jumping straight to the solution. This reduces the chance of making mistakes.

  3. Incorrect Graphing: Make sure to use open circles for inequalities that do not include equal (e.g., \( < \)) and closed circles for those that do (e.g., \( \leq \)).

Additional Practice

Practice is key to mastering two-step inequalities. Here are some examples for you to try:

  1. Solve and graph: \( 4x + 10 > 26 \)
  2. Solve and graph: \( -5x + 3 \leq 18 \)
  3. Solve and graph: \( 2x - 8 < 6 \)

Conclusion

Solving two-step inequalities is a fundamental skill in algebra that lays the groundwork for understanding more complex inequalities. Remember, the key steps are to isolate the variable, reverse directions when necessary, and graph your solutions accurately. With practice, you'll find that solving these inequalities becomes a straightforward and rewarding task. Happy solving!