Study Guide

Equations and Inequalities - Single-Variable Inequalities

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Single-Variable Inequalities

An equation is a statement that two quantities are equal. An inequality is a statement comparing two quantities that may or may not be equal. As we can compare one real number to another, we can compare a real number and a variable. This may seem a little like comparing apples and oranges, but is it really that hard to compare apples and oranges? Apples are redder, oranges have more stuff to peel off. There, done.

We can represent inequalities using words, symbols, or pictures on a number line.

  • Strict Inequalities

    These inequalities don't take no guff. They lay down the law. Better not cross 'em.

    The symbol "<" abbreviates "is less than" while the symbol ">" abbreviates "is greater than." We personally think the latter symbol is superior, which we can express with this intentionally confusing series of symbols: "> > <." Pretty cool, right? Right?

    When using the symbols "<" and ">," the larger quantity is on the big, open side of the symbol, while the smaller quantity gets the little tiny point of the symbol. If it helps, picture the lines of the symbol extending outward so the value on the open side dwarfs the one on the pointy side.

    Don't forget that this symbol can be flip-flopped so it's not pointing in the same direction every time. Always check for the big open side. This little guy is hungry; he's always trying to get his chompers on the biggest meal.

    Sample Problems

    1. "Three is less than x" can be written in symbols as 3 < x
    2. We can abbreviate "x is less than -1" by x < -1.
    3. "Four is greater than y" can be written "4 > y".
    4. The statement in symbols "> 100" means "x is greater than 100."

    A solution of an inequality is any number that satisfies the inequality, or makes the inequality true. It's all about satisfaction. The inequality will be bummed if it can't get no satisfaction. Um, solution.

    Remember that as we go further to the right on the number line, the numbers become larger, and as we go further left, the numbers become smaller. They may get even smaller yet if they take a sip from that bottle with the label "Drink Me."

    Sample Problems

    1. x = 4 is a solution to the inequality 3 < x, since 4 is to the right of 3 on the number line.
    2. x = -2 satisfies the inequality x < - 1, so x = - 2 is a solution to this inequality.
    3. y = 4 is not a solution to the inequality 4 > y, because 4 is not greater than itself.
    4. x = 5 is not a solution to the inequality x > 100. Don't even bother trying to pay for that $100 digital camera with a $5 bill. Trust us.

    By the way, inequalities that use the "<" and ">" symbols are also known as strict inequalities because they're super strict about excluding the value next to them. For example, x > 9 includes all values greater than 9, but not 9 itself. It's strict like that.

  • Equivalent Inequalities

    Equivalent inequalities are inequalities with the same solutions. They're 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.

    Sample Problem

    Are the inequalities x > 3 and  3 < x equivalent?

    They both say that x must be larger than 3. No bickering here. So yep, they're equivalent.

    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, 2 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 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're 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.

    Sample Problem

    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.

    Sample Problem

    To represent the inequality 4 > x or, equivalently, the inequality x < 4, we shade all values up to but not including 4. This time we're showing the opposite end of the number line some love.

  • Non-Strict Inequalities

    Inequalities that Allow Equality

    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's 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 "≥". The "≤" symbol 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

    1. "Two is less than or equal to x" can be written in symbols as 2 ≤ x. In other words, x must be at least 2.
    2. We can abbreviate "x is less than or equal to -1" as x ≤ -1. 
    3. The inequality 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's being compared. These guys are known in the math world as non-strict inequalities. We might also 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's 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

    1. y = 4 is a solution to the inequality 4 ≥ y.
    2. x = 5 is a solution to the inequality x ≥ 5.

    We still have equivalent ways to write non-strict 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 non-strict inequality on the number line, use a closed circle or a big filled-in 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.

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