Chapter 2, Section 6: Inequalities - Solving and Graphing

Welcome to Chapter 2, Section 6, where we'll be focusing on inequalities! This section builds upon your knowledge of equations and introduces the concepts needed to understand, solve, and graph inequalities. Get ready to expand your mathematical toolkit!

What are Inequalities?

Unlike equations, which state that two expressions are equal, inequalities compare expressions using symbols like:

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

For example, $x < 5$ means that $x$ can be any number smaller than 5, but not equal to 5. On the other hand, $x \leq 5$ means $x$ can be any number smaller than or equal to 5.

Solving Inequalities

Solving inequalities is very similar to solving equations, with one crucial difference: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign!

Here's a step-by-step approach:

  1. Simplify both sides of the inequality by combining like terms and using the distributive property.
  2. Isolate the variable term. Add or subtract the same value from both sides to get the variable term alone on one side.
  3. Isolate the variable. Multiply or divide both sides by the coefficient of the variable. Remember to flip the inequality sign if you multiply or divide by a negative number!

Example:

Let's solve the inequality $ -2x + 3 > 7 $

  1. Subtract 3 from both sides: $ -2x > 4 $
  2. Divide both sides by -2 (and flip the inequality sign!): $ x < -2 $

Therefore, the solution to the inequality is $ x < -2 $. This means any number less than -2 will satisfy the original inequality.

Graphing Inequalities on a Number Line

Visualizing solutions to inequalities is easiest by graphing them on a number line. Here's how:

  • Draw a number line.
  • Locate the boundary point. This is the number that $x$ is being compared to (e.g., -2 in the example above).
  • Draw a circle at the boundary point. Use an open circle (o) if the inequality is $<$ or $>$, indicating the boundary point is not included in the solution. Use a closed circle (•) if the inequality is $\leq$ or $\geq$, indicating the boundary point is included.
  • Shade the number line to the left or right of the boundary point, depending on the direction of the inequality. Shade to the left for $<$ or $\leq$, and to the right for $>$ or $\geq$.

So, the graph of $x < -2$ would have an open circle at -2 and the line shaded to the left, indicating all numbers less than -2.

Key Takeaways

  • Solving inequalities is similar to solving equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
  • Graphing inequalities on a number line provides a visual representation of the solution set.

Keep practicing these concepts, and you'll master inequalities in no time! Good luck!