Solving and Graphing Linear Inequalities

1. From Equality to Inequality

An equation uses an equals sign (=) to state that two expressions are identical in value. An inequality uses one of four symbols to compare expressions:

• < : less than
• > : greater than
• ≤ : less than or equal to
• ≥ : greater than or equal to

The solution to a linear equation like $x+3=5$ is a single point ($x=2$). The solution to an inequality like $x+3>5$ is a range of values (in this case, all numbers greater than 2). We often represent this range visually on a number line.

This image provides an excellent overview of how inequalities are represented.

The open circle (like for $x>1$) means the number 1 is not included in the solution. The closed circle (like for $x\le 2$) means the number 2 is included in the solution.

2. The Rules for Solving Inequalities

To solve an inequality, you use the same inverse operations (addition/subtraction, multiplication/division) to isolate the variable, just as you did with equations. There is, however, one critical exception.

Let's read a short document that clearly lays out the rules of manipulation.

The Golden Rule of Inequalities

As you just read, the process is the same as for equations, except for one situation.

Key Rule: If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.

Let's see why. We know that $5>2$. This is true.
But if we multiply both sides by $-1$, we get $-5$ and $-2$.
Is $-5>-2$? No. On a number line, $-5$ is to the left of $-2$, so it's smaller. We must flip the sign to make the statement true: $-5<-2$.

Example: Solve $-3x+4\le 16$

1. Subtract 4 from both sides:
   $-3x\le 12$
2. Divide both sides by -3. Because we are dividing by a negative number, we must flip the inequality sign from $\le$ to $\ge$.
   $\frac{-3x}{-3}\ge \frac{12}{-3}$
   $x\ge -4$

The solution is all numbers greater than or equal to -4. On a number line, this would be a closed circle at -4 with an arrow pointing to the right.

3. A Comprehensive Video Walkthrough

Now, let's watch a video that brings all these concepts together. It uses British terminology and covers a range of examples, from basic problems to more complex ones.

Key Concepts from the Video

• Solving with Variables on Both Sides: The video reinforces the strategy you learned for equations: collect the variable terms on one side (preferably by eliminating the smaller one) and the constant terms on the other.
• Compound Inequalities: For an inequality like $-5\le 2x+3<7$, whatever operation you do to isolate $x$ in the middle, you must also do to both the left and right sides.
• Number Lines: The video provides great practice for representing solutions on a number line, including the conventions for open/closed circles and compound inequalities.

Conclusion

Well done! You've successfully extended your algebra skills from equations to inequalities. The process is remarkably similar, which shows how powerful these core principles are.

Here are the key takeaways from today's lesson:

• Solving Process: Use inverse operations to isolate the variable, just as you would with an equation.
• The Golden Rule: When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the inequality symbol (e.g., < becomes >).
• Number Line Representation:
  • Use an open circle for < and > to show the endpoint is not included.
  • Use a closed (filled-in) circle for ≤ and ≥ to show the endpoint is included.

In our next lesson, we will venture into problems with two variables by learning how to solve systems of two linear equations. You'll discover methods to find a pair of values ($x,y$) that satisfy two different equations at the same time.