Solving One-Step Inequalities

When working on algebraic problems, one of the fundamental skills you will encounter is how to solve inequalities. Understanding how to solve one-step inequalities is a crucial step in your mathematical journey, as it lays the groundwork for more complex concepts down the line. So, let’s dive right into the methods, examples, and visual representations involved in solving one-step inequalities effectively!

What is an Inequality?

An inequality is a mathematical statement that compares two expressions using symbols like greater than \(>\), less than \(<\), greater than or equal to \(\geq\), and less than or equal to \(\leq\). While equations tell us that two expressions are equivalent, inequalities show us that one expression can be larger or smaller than another.

Understanding One-Step Inequalities

One-step inequalities require only a single operation to isolate the variable. The goal is to find the solution that makes the inequality true. Let's explore the steps used to solve one-step inequalities!

Step 1: Identify the Inequality

First, identify the type of inequality you are dealing with:

  • Greater than (>): This means the left-hand side is larger than the right-hand side.
  • Less than (<): This indicates that the left-hand side is smaller than the right-hand side.
  • Greater than or equal to (≥): This indicates that the left side is larger than or exactly equal to the right side.
  • Less than or equal to (≤): This signifies that the left side is smaller than or exactly equal to the right side.

Step 2: Solve the Inequality

To solve a one-step inequality, you'll use basic arithmetic operations: addition, subtraction, multiplication, or division. The key point to remember is that if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Example 1: Solving a One-Step Inequality with Addition

Let’s look at an example:

\[ x + 5 < 12 \]

Step 1: Isolate the variable by subtracting 5 from both sides:

\[ x + 5 - 5 < 12 - 5 \]

\[ x < 7 \]

The solution \(x < 7\) tells us that \(x\) must be less than 7.

Example 2: Solving a One-Step Inequality with Subtraction

Now, let’s try one with subtraction:

\[ y - 3 \geq 4 \]

Step 1: Add 3 to both sides to isolate \(y\):

\[ y - 3 + 3 \geq 4 + 3 \]

\[ y \geq 7 \]

This solution, \(y \geq 7\), means \(y\) can be 7 or any number larger than 7.

Example 3: Solving a One-Step Inequality with Multiplication

Here’s another example:

\[ 3z > 15 \]

Step 1: Divide both sides by 3 to solve for \(z\):

\[ \frac{3z}{3} > \frac{15}{3} \]

\[ z > 5 \]

This solution indicates that \(z\) must be greater than 5.

Example 4: Solving a One-Step Inequality with Division by a Negative Number

Let’s see how the sign flips:

\[ -2m \leq -10 \]

Step 1: Divide both sides by -2. Remember to flip the inequality sign:

\[ \frac{-2m}{-2} \geq \frac{-10}{-2} \]

\[ m \geq 5 \]

This indicates that \(m\) must be greater than or equal to 5.

Step 3: Representing Solutions on a Number Line

Once you've found the solution to an inequality, it's helpful to visually represent it on a number line. This can reinforce understanding and provide a clear reference.

Example Representation

  • For the solution \(x < 7\):

    • Draw an open circle on 7 (indicating that 7 is not included).
    • Shade to the left, indicating all numbers less than 7.

  • For \(y \geq 7\):

    • Draw a closed circle on 7 (indicating that 7 is included).
    • Shade to the right, indicating all numbers greater than or equal to 7.

Why Is This Important?

Understanding how to solve one-step inequalities is essential not just for academic purposes, but it applies to real-world scenarios as well! Whether you’re budgeting, assessing risks, or making decisions, inequalities can provide valuable insights.

Practice Problems

To really cement your understanding, here are a few practice problems:

  1. Solve the inequality: \(x - 4 < 10\)

  2. Solve the inequality: \(3y \geq 9\)

  3. Solve the inequality: \(-5z < -25\)

  4. Solve the inequality: \(8 + w \leq 12\)

Answers to Practice Problems

  1. \(x < 14\)
  2. \(y \geq 3\)
  3. \(z > 5\)
  4. \(w \leq 4\)

Conclusion

Now that you have a solid understanding of how to solve one-step inequalities, along with representing solutions on a number line, you’re well-equipped to tackle even more challenging problems in algebra. Keep practicing, and remember that mastering these concepts is crucial for your success in mathematics!

By continuing to refine your skills, you’ll discover the beauty of inequalities and how they facilitate a deeper understanding of relationships between quantities. Happy solving!