Study Guide

Word Problems - Equations and Inequalities

Equations and Inequalities

English has nouns (cat, fish, diaper) and verbs (scratch, swim, poop). A proper English sentence contains at least one noun and one verb ("The cat ate.''). Hopefully, the cat did not eat poop. Again.

In algebra, expressions act much like nouns with the verbs being relations, which are symbols that say how one quantity relates to another, such as =, <, ≥, and so on. An algebraic sentence is a statement of relation between two quantities, such as 2x = 4, which could be read as "two times x equals 4." Put into a phrase that sounds like something you might actually hear, we could say, for example, "I have 4 Christmas presents, which is 2 times as many as Xavier got." We wouldn't brag about that though. Sounds like you both got shafted.

Now that we've got the hang of expressions, it's time to put them together into sentences. So to speak.

  • The Equal Sign

    Hey equation, what's your sign?

    There are many different English phrases for each of the arithmetic operators, and there are likewise many different English phrases that translate into the = sign. In fact, "likewise" is one word that roughly means = . Any English phrase that says two quantities are the same, or will be the same if you perform some operations first, translates into a mathematical equation using the = sign. For example, James Bond may tell you that the world = not enough.

    Sample Problems

    • "x plus 2 equals 10'' translates into symbols as x + 2 = 10.
       
    • "Three x is twenty-four'' translates into symbols as 3x = 24.
       
    • "One-half of y will be seven'' translates to .
       
    • "If we divide y by four we get twelve'' translates to .
       
    • "Multiplying z by two gives (or yields) eleven'' translates to z × 2 = 11.
  • Inequalities

     

    We have four possible inequality signs to choose from: <, >, ≤, and ≥. There's also the slightly less popular ^, which means that everything below is greater than everything above, but that brings us into a touchy area, spiritually speaking. Translating English into one of these symbols is a matter of common sense. Good thing you have some of that.

    The signs < and > are usually straightforward.

    • "Three is less than x'' can be abbreviated by 3 < x.
       
    • "5 is more than x'' can be written in symbols as 5 > x.

    When we're translating into statements that use ≤ or ≥, we have a slightly wider range of English phrases to use. For example, each of the following English statements translates into symbols as x ≤ 20.

    • x is less than or equal to twenty.
       
    • x is at most twenty.
       
    • x is not more than twenty.
       
    • x is not greater than twenty.
       
    • Twenty could be more than x, or it could be exactly as much as x. You'll never know. Na-nana-boo-boo.

    Each of these following English statements translates into symbols as x ≥ 0.

    • x is greater than or equal to zero.
       
    • x is not less than zero.
       
    • x is not smaller than zero.
       
    • x is at least zero.

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