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Using Special Symbols

Sometimes SAT writes equations using symbols that aren’t actually math symbols.  They use stars or hearts or other strange symbols.  The symbols have no intrinsic meaning; you just do what the problem tells you to do by following the example. 

Sometimes special symbols are just “blanks.”  You just have to replace the symbol with a number that works. 

Example:

           $\text{If }6+\bigtriangleup=18\text{, what is }\bigtriangleup\text{?}$

Just replace the $\bigtriangleup$ with a number that works.

             $6 + 12 = 18\text{, so } \bigtriangleup=12$

Sometimes special symbols stand for operations.  These kinds of special symbol problems ALWAYS include an example showing you how to do the problem. 

To do special symbol problems, look at the problem and look at the example.  Take the numbers from your problem and arrange them in the EXACT SAME way that they are arranged in the example problem.  Many students get confused because they don't know what the symbols "mean."  They don't mean anything more than what the problem tells you.  The example shows you how to arrange your numbers. Arrange them correctly, do the math, and you'll have your answer.

Example:

$7\clubsuit3=7+3$

$\text{What is } 5\clubsuit6\text{?}$

           In this case, you know that numbers on either side of a $\clubsuit$ are supposed to be added together, so add $5+6=11$.

Other problems are a little more complicated:

Example:

$x\bigstar a=x(a)+(x-a)$

$\text{Find } 3\bigstar 2\text{.}$

In a problem like this, you look closely at how variables (or numbers) are arranged around the symbol, and then how they are arranged in the equation.  

In the example, $x$ is in front of $\bigstar$ and in the bolded positions in the equation: $\mathbf{x}(a)+(\mathbf{x}-a)$.  For our problem, $3$ is in front of $\bigstar$ so $3$ goes into those same positions:

$\mathbf{3}(a)+(\mathbf{3}-a)$

In the example, $a$ is behind $\bigstar$ and in the bolded positions in the equation: $x(\mathbf{a})+(x-\mathbf{a})$.  For our problem, $2$ is behind $\bigstar$ so $2$ goes into those same positions:

$x(\mathbf{2})+(x-\mathbf{2})$

For a final equation of:

$$\eqalign{3(2)+(3-2)&=\\6+1&=7}$$

Tests love special symbols because they look hard.  If you think of them as puzzles, in which you just have to make sure you get the right pieces in the right places, they aren't difficult at all. 

Practice Problems:

  • Special Symbols

    Find the value of the special symbol:

    1. If $3+\triangle=7$ what is the value of $\triangle$?
    2. If $2(\star + 9)=12$ what is the value of $\star$?
    3. If $2\diamond+4 - 3\diamond=2 + \diamond$, what is the value of $\diamond$?
    4. If $\dfrac{\circledcirc}{3}-11=\dfrac{1}{2}$ what is the value of $\circledcirc$?
    5. If $\dfrac{1}{\square}+8=-10$ what is the value of $\square$?

    Using the rules provided by the equation with the special symbols, find the value of the following expressions:

     

    $x\circledS y\circledS z = x+(y)(z)$

    1. $3\circledS 4\circledS 5 $
    2. $1\circledS 2\circledS 10 $
    3. $12\circledS 2\circledS 3 $

    $x\spadesuit y\spadesuit z = z(x^y)$

    1. $1\spadesuit 2\spadesuit 3$
    2. $5\spadesuit 1\spadesuit 8$
    3. $3\spadesuit 0\spadesuit 100$

    $\dfrac{\heartsuit}{x}=\sqrt{x}+7$

    1. $\dfrac{\heartsuit}{4}$
    2. $\dfrac{\heartsuit}{64}$
    3. $\dfrac{\heartsuit}{10}$

    Answer Key:

Test Prep Practice

  • Algebra: Using Special Symbols

    1. Let $x$ § $y = x ^y +y$ for all positive integers of $x$ and $y$. What is the value of $4$ § $2$?

    (A) 20

    (B) 18

    (C) 16

    (D) 22

    (E) 24

     

     

    2. ¤ • 2 + 8 ÷ 4 = 34

    What number when used in place of ¤ above, makes the statement true?

    (A) 16

    (B) 28

    (C) 42

    (D) 60

    (E) 64

     

    3. Let $a$ Ω $b$ = $(-2a)^3 + b^2$ for all positive integers $a$ and $b$.

    State the value of $3$ Ω $12$.

    (A) 360

    (B) -360

    (C) 72

    (D) -72

    (E) -198

     

     

    4. Let $x$ ♠ $y$ ♠ $z$ = 5 ($x$ + $z$) - $y^2$ for all positive integers of $x$, $y$, and $z$. What is the value of 7 ♠ 8 ♠ 9?

     

     

Common Core Grade Level/Subject