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SAT Math : Algebra

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Example Questions

Example Question #12 : Algebraic Fractions

Which of the following provides the complete solution set for |2-z|>0 ?

Possible Answers:

No solutions

z>2

z<2

z=2

$\dpi{100} z|z\neq 2$

Correct answer:

$\dpi{100} z|z\neq 2$

Explanation:

The absolute value will always be positive or 0, therefore all values of z will create a true statement as long as $2-z\neq 0$ . Thus all values except for 2 will work.

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Example Question #1 : How To Find Excluded Values

If the average (arithmetic mean) of , , and is , what is the average of x+2 , y-6 , and ?

Possible Answers:

There is not enough information to determine the answer.

Correct answer:

Explanation:

If we can find the sum of $\dpi{100} \small x+2$ , $\dpi{100} \small y-6$ , and 10, we can determine their average. There is not enough information to solve for $\dpi{100} \small x$ or $\dpi{100} \small y$ individually, but we can find their sum, $\dpi{100} \small x+y$ .

Write out the average formula for the original three quantities. Remember, adding together and dividing by the number of quantities gives the average: $\frac{x + y + 9}{3} = 12$

Isolate $\dpi{100} \small x+y$ :

$x + y + 9 = 36$

$x + y = 27$

Write out the average formula for the new three quantities:

$\frac{x + 2 + y - 6 + 10}{3} = ?$

Combine the integers in the numerator:

$\frac{x + y + 6}{3} = ?$

Replace $\dpi{100} \small x+y$ with 27:

$\frac{27+ 6}{3} = \frac{33}{3} = 11$

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Example Question #2 : How To Find Excluded Values

Find the excluded values of the following algebraic fraction

$\frac{2x^2-5x-12}{x^2-11x+28}$

Possible Answers:

The numerator cancels all the binomials in the denomniator so ther are no excluded values.

$x\neq4,7$

$x\neq4$

$x\neq7$

$x\neq-11$

Correct answer:

$x\neq4,7$

Explanation:

To find the excluded values of a algebraic fraction you need to find when the denominator is zero. To find when the denominator is zero you need to factor it. This denominator factors into

(x-7)(x-4)

so this is zero when x=4,7 so our answer is

$x\neq4,7$

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Example Question #2 : How To Find Excluded Values

$y=\frac{x^2-5}{x^2-25}$

For what value(s) of x is the function undefined?

Possible Answers:

x=5, x=-5, x=0

x=-5

$x=-5, x=\sqrt{5}$

$x=\sqrt{5}$

x=5, x=-5

Correct answer:

x=5, x=-5

Explanation:

When the denominator of a function is equal to 0, the function is undefined at that point. We can set x²-25 equal to 0 in order to find out what values of x make that true.

x^2-25=0

We can factor to solve for x.

(x+5)(x-5)=0

x+5=0, x-5=0

x=5, x=-5

$x=\sqrt{5}$ is an incorrect answer because for this value of x, the function equals zero, but it is not undefined.

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Example Question #2 : How To Find Excluded Values

Find the extraneous solution for

$\sqrt{x}=3x-4$

Possible Answers:

There are no extraneous solutions

x=-1

$x=\frac{16}{9}$

x=1

Correct answer:

x=1

Explanation:

$\sqrt{x}=3x-4$

$(\sqrt{x})^2=(3x-4)^2$

x=9x^2-24x+16

0=9x^2-25x+16

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{25\pm\sqrt{(-25)^2-4\cdot9\cdot16}}{2\cdot9}$

$x=\frac{25\pm\sqrt{625-576}}{18}$

$x=\frac{25\pm\sqrt{49}}{18}$

$x=\frac{25+7}{18}=\frac{32}{18}=\frac{16}{9}$

$x=\frac{25-7}{18}=\frac{18}{18}=1$

Now lets plug these values into our original equation.

For $x=\frac{16}{9}$

$\sqrt{x}=3x-4$

$\sqrt{\frac{16}{9}}=3\left(\frac{16}{9}\right)-4$

$\frac{4}{3}=\frac{16}{3}-4$

$\frac{4}{3}=\frac{16}{3}-\frac{12}{3}=\frac{4}{3}$

So this isn't an extraneous solution.

For x=1

$\sqrt{x}=3x-4$

$\sqrt{1}=3(1)-4$

1=3-4=-1

Since $1\neq-1$ , x=1 is an extraneous solution.

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Example Question #1 : Distributive Property

$(\sqrt5+\sqrt3)^2=$

Possible Answers:

$8+2\sqrt{15 }$

$2+\sqrt{15}$

$\sqrt{15}$

$8+\sqrt{15}$

$2+2\sqrt{15}$

Correct answer:

$8+2\sqrt{15 }$

Explanation:

$(\sqrt5+\sqrt3)^2=(\sqrt5)^2+2\sqrt5\sqrt3+(\sqrt3)^2=5+2\sqrt{15}+3=8+2\sqrt{15}$

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Example Question #1 : How To Use Foil In The Distributive Property

If (3x - 4)(3x + 4) = 2 , what is the value of 9x^2 ?

Possible Answers:

Correct answer:

Explanation:

Remember that (a – b )(a + b ) = a ²– b ².

We can therefore rewrite (3x – 4)(3x + 4) = 2 as (3x )²– (4)²= 2.

Simplify to find 9x²– 16 = 2.

Adding 16 to each side gives us 9x²= 18.

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Example Question #2 : How To Use Foil In The Distributive Property

If g(x)=2x^2-2 and h(x)=x+4 , then which of the following is equivalent to g(h(x))-h(g(x)) ?

Possible Answers:

-16x-28

2x^3+8x^2-2x-8

16x

16x+28

Correct answer:

16x+28

Explanation:

We are asked to find the difference between g(h(x)) and h(g(x)), where g(x) = 2x² – 2 and h(x) = x + 4. Let's find expressions for both.

g(h(x)) = g(x + 4) = 2(x + 4)² – 2

g(h(x)) = 2(x + 4)(x + 4) – 2

In order to find (x+4)(x+4) we can use the FOIL method.

(x + 4)(x + 4) = x² + 4x + 4x + 16

g(h(x)) = 2(x² + 4x + 4x + 16) – 2

g(h(x)) = 2(x² + 8x + 16) – 2

Distribute and simplify.

g(h(x)) = 2x² + 16x + 32 – 2

g(h(x)) = 2x² + 16x + 30

Now, we need to find h(g(x)).

h(g(x)) = h(2x² – 2) = 2x² – 2 + 4

h(g(x)) = 2x² + 2

Finally, we can find g(h(x)) – h(g(x)).

g(h(x)) – h(g(x)) = 2x² + 16x + 30 – (2x² + 2)

= 2x² + 16x + 30 – 2x² – 2

= 16x + 28

The answer is 16x + 28.

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Example Question #2 : How To Use Foil In The Distributive Property

The sum of two numbers is . The product of the same two numbers is . If the two numbers are each increased by one, the new product is . Find q-p in terms of .

Possible Answers:

s+1

s^2

s-1

s+2

2s+1

Correct answer:

s+1

Explanation:

Let the two numbers be x and y.

x + y = s

xy = p

(x + 1)(y + 1) = q

Expand the last equation:

xy + x + y + 1 = q

Note that both of the first two equations can be substituted into this new equation:

p + s + 1 = q

Solve this equation for q – p by subtracting p from both sides:

s + 1 = q – p

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Example Question #5 : Distributive Property

Expand the expression:

$\dpi{100} \small (x^{3}-4x)(6 + 12x^{2})$

Possible Answers:

$\dpi{100} \small 12x^{5}-42x^{3}-24x$

$\dpi{100} \small 6x^{3} + 12x^{5}-24x-48x^{3}$

$\dpi{100} \small 22x^{2}$

$\dpi{100} \small 42x^{3}+12x^{5}-24x$

$\dpi{100} \small 6x^{3} + 12x^{2}-24x-48$

Correct answer:

$\dpi{100} \small 12x^{5}-42x^{3}-24x$

Explanation:

When using FOIL, multiply the first, outside, inside, then last expressions; then combine like terms.

$\dpi{100} \small (x^{3}-4x)(6 + 12x^{2})$

$\dpi{100} \small 6x^{3}+12x^{5}-24x-48x^{3}$

$\dpi{100} \small -42x^{3}+12x^{5}-24x$

$\dpi{100} \small 12x^{5}-42x^{3}-24x$

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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #12 : Algebraic Fractions

Example Question #1 : How To Find Excluded Values

Example Question #2 : How To Find Excluded Values

Example Question #2 : How To Find Excluded Values

Example Question #2 : How To Find Excluded Values

Example Question #1 : Distributive Property

Example Question #1 : How To Use Foil In The Distributive Property

Example Question #2 : How To Use Foil In The Distributive Property

Example Question #2 : How To Use Foil In The Distributive Property

Example Question #5 : Distributive Property

All SAT Math Resources

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