Simple Inequalities - 9/16/13

Welcome to Professor Baker's Math Class! Today, we're diving into the world of inequalities. Inequalities help us describe situations where values are not exactly equal but rather greater than, less than, or within a certain range. Let's explore how to solve and graph them!

What are Inequalities?

An inequality is a mathematical statement that compares two expressions using inequality symbols. Here are some common symbols:

  • $<$ : Less than
  • $>$ : Greater than
  • $\le$ : Less than or equal to
  • $\ge$ : Greater than or equal to
  • $\neq$ : Not equal to

Solving Inequalities

Solving inequalities is similar to solving equations, but with one crucial difference: when you multiply or divide by a negative number, you must reverse the inequality sign. For example:

Solve for $x$: $3x + 2 \ge -4$

  1. Subtract 2 from both sides: $3x \ge -6$
  2. Divide both sides by 3: $x \ge -2$

So the solution is $x \ge -2$

Graphing Inequalities

We can represent the solutions of inequalities on a number line. Here's how:

  • Open Dot ($\circ$): Used for strict inequalities ($<$ or $>$) to indicate that the endpoint is not included in the solution.
  • Closed Dot ($\bullet$): Used for inequalities that include equality ($\le$ or $\ge$) to indicate that the endpoint is included in the solution.

For example:

  • $x > -2$ is represented with an open dot at -2 and an arrow extending to the right.
  • $x \le 3$ is represented with a closed dot at 3 and an arrow extending to the left.

Let's look at a compound inequality. For example, $-3 \le x < 2$ is graphed with a closed dot at -3 (because of the $\le$) and an open dot at 2 (because of the $<$), with a line connecting the two dots to show that $x$ is between -3 and 2.

Example Problem

Let's say we have a distance problem: A train travels at a speed of 32 km/h and covers a distance of 547 km. How long does it take?

Using the formula Distance = Rate $\times$ Time, or $D = RT$, we have:

$547 = 32T$

Dividing both sides by 32, we get:

$T = \frac{547}{32} \approx 17.1$ hours.

Homework

Remember to practice what we learned today! Complete the following problems to solidify your understanding:

Page 45, Problems #13-36 (all)

Keep practicing, and you'll master inequalities in no time! Good luck, and see you in the next class!