Equations and Inequalities
Topics
Introduction to :
Okay, time out. How can you be an inequality and allow equality? Isn't that an oxymoron? Can you also perform an injustice that is just? Or have indigestion while digesting?
The answer is that some inequalities are not entirely unequal: there's an overlap. The inequalities we've already worked with—the ones involving the symbols "<" and ">"—are called strict inequalities, because the variable isn't allowed to equal the number to which it is being compared. If x < 3 then x can be 2.9, 2.99, 2.999, and so on, but x can't equal 3.
However, we can also write inequalities involving the symbols "≤" and "≥". "≤" means "less than or equal to," while "≥" is an abbreviation for "greater than or equal to." The values on each side of the symbol aren't exactly equal, making it an inequality, and yet one of the possible solutions does equal the value on the opposite side, therefore making it slightly equal. Got that?
Sample Problems
- "Two is less than or equal to x" can be written in symbols as 2 ≤ x. x must be at least 2.
- We can abbreviate "x is less than or equal to - 1" as x ≤ - 1.
- "4 ≥ y" means "4 is greater than or equal to y". In other words, y is at most 4.
The inequalities "≤" and "≥" allow the variable to equal the number to which it is being compared. There isn't a special name for these inequalities like there is for strict inequalities. We feel bad about that, so we'll make one up right now. We'll call them "lenient inequalities." Hey, that's actually good. That might catch on.
Since "≤" and "≥" allow the variable to equal the number to which it is being compared, we can think of them as inequalities that "allow equality," or as "relaxed" inequalities. We like the second one. It makes us think of vacation.
Sample Problems
- y = 4 is a solution to the inequality 4 ≥ y.
- x = 5 is a solution to the inequality x ≥ 5.
We still have equivalent ways to write relaxed inequalities.
Sample Problem
To represent the inequality 4 ≥ x or the equivalent inequality x ≤ 4, we shade all values up to and including 4.
Be Careful: To represent a strict inequality on a number line, use an empty circle. To represent a relaxed inequality on the number line, use a big dot. Be careful to draw the right kind of circle and pay close attention to the pictures that you get. If you get a landscape of the Swiss countryside, something has gone awry somewhere along the line.
Example 1
x ≥ 3 and 3 ≤ x are equivalent. They both say that x must be at least 3. Demonstrate this. |
Example 2
To represent the inequality 3 ≤ x we draw a number line labeled with the name of the variable, and put a big dot at 3: |
Exercise 1
Write the statement as an inequality using ≤ or ≥ as appropriate: 5 is less than or equal to x.
Exercise 2
Write the statement as an inequality using ≤ or ≥ as appropriate: -10 is greater than or equal to x.
Exercise 3
Write the statement as an inequality using ≤ or ≥ as appropriate: y is at most 3.
Exercise 4
Write the statement as an inequality using ≤ or ≥ as appropriate: z is at least 1.
Exercise 5
For the picture, what two equivalent inequalities are represented?
Exercise 6
For the picture, what two equivalent inequalities are represented?
Exercise 7
For the picture, what two equivalent inequalities are represented?
Exercise 8
For the picture, what two equivalent inequalities are represented?
Exercise 9
Fill in the following table. The first row is filled in as an example. Pay careful attention to whether inequalities are strict or relaxed. If strict, maybe offer them a chill pill?