Equations and Inequalities
Equivalent inequalities are inequalities with the same solutions. They are also the most likely to get paired up together on eHarmony. It's incredible how much they have in common. It's like they were made for each other.
x > 3 and 3 < x are equivalent. They both say that x must be larger than 3. No bickering here.
Inequalities usually have a lot of solutions—in fact, infinitely many. Think about the inequality x > 3. This inequality states that "x must be larger than 3." Any number bigger than 3 is a solution to this inequality. That includes 3.001, 3.0001, 4, 5, two million, and every other number bigger than 3. We don't have time at the moment to name them all, but let's schedule something for next week. The nicest way to write the solutions to the inequality is to write x > 3 again, but this time think of x > 3 as meaning "the set of all real numbers greater than 3." That's a fairly big set. Like, bigger than the ones they used to shoot the Pirates of the Caribbean movies.
A not-so-nice way to picture the inequality?
Another way to show the solutions to an inequality, or to represent an inequality itself, is to use a number line. Wind up your number line, toss it out into the depths, and reel yourself in an inequality. Okay, so it doesn't really work that way, but is anyone else suddenly in the mood to go fishing? No? Just us?
We start out by labeling a number line with the name of the variable we are working with. Then we shade in or color the values of the variable that are solutions to the equation. Yes, you may use pink. We can think of the picture as showing the inequality, or as showing the solutions to the inequality.
To represent the inequality 3 < x, we first draw a number line and label it with the name of the variable:
We draw an empty circle around 3 to show that x cannot equal 3, even though it can come awfully close:
Then we shade all values on the number line greater than (to the right of) 3, since these values are the solutions to the inequality. Go ahead—don't be afraid to make your number line extra shady.
To represent the inequality 4 > x or, equivalently, the inequality x < 4, we shade all values up to but not including 4. Here, we are showing the opposite end of the line some love.