These questions typically appear early in the math sections, and only need the 4 basic operations (add, subtract, multiply, divide) to solve them.

Examples:

$$x-7=12$$

$$4x+3=15$$

$$\frac x6+\frac23=4$$

Per usual on the SAT, they will sometimes try and make these look harder by introducing another variable or constant. Don’t be fooled – you’re just solving for *x*. They key for these is any time you are given an additional value, immediately plug it in.

Examples:

$$\sqrt{x+a}-3=1$$

If *a* = 1, what is the solution set of the equation above?

If \(x=\frac34y\) and \(y=16\), what is the value of \(3x+2\)?

**Strategy:**

**Perform your order of operations correctly. You may have learned the acronym PEMDAS for order of operations (if you didn’t, now is a good time!). Remember, it stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.****If you have a square root in the original problem, ALWAYS test for extraneous solutions. 95% of the time you will have one. If you have a fraction in the original equation, always check that you are not dividing by 0.**

__Manipulating Polynomials__

The last type of “basic” algebra question you will see requires you to know how to manipulate polynomials. These are the same rules as the single-value equations, but generally require you to deal with exponents in some manner.

A common question:

Which expression is equivalent to $$\left(3x^3-5\right)-3(x^3-x^2-2)$$

A) \(6x^3+3x^2-7\)

B) \(-x^2-7\)

C) \(3x^2+1\)

D) \(3x^3-3\)

**Strategy:**

**Perform your order of operations correctly.****Combine like terms (same variable and exponent)****Be very careful distributing negatives!**

In our example on the previous page, they are testing three skills – do you know your order of operations, do you know how to combine like terms, and do you know how to distribute both multiplication AND a +/- sign.

Here, we will first need to distribute our -3. Then, we will need to combine like terms (addition and subtraction fall last).

Doing so will bring us to answer C).

Another example the SAT regards as very difficult:

Which of the following is equivalent to \(\left(a+\frac b4\right)^2\)?

A) \(a^2+\frac{b^2}4\)

B) \(a^2+\frac{b^2}{16}\)

C) \(a^2+\frac{ab}4+\frac{b^2}4\)

D) \(a^2+\frac{ab}2+\frac{b^2}{16}\)

When presented with something like this, DO NOT just distribute the exponent! Notice the answer choices are tempting you to do just that. Re-write the problem like:

$$\left(a+\frac b4\right)\left(a+\frac b4\right)$$

and then FOIL to get:

$$a^2+\frac{ab}4+\frac{ab}4+\frac{b^2}{16}$$

we then combine our like terms, reduce the middle fraction, and are left with answer D).

Students miss these problems all the time due to the reasons listed above: not checking for extraneous solutions, not applying exponent rules correctly, or not keeping close enough track of +/- signs. If you follow all the algebra rules, these problems are very straightforward.

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