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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #6 : How To Find The Solution To An Inequality With Multiplication

Fill in the circle with either $<$ , $>$ , or $=$ symbols:

$(x-3)\circ\frac{x^2-9}{x+3}$ for $x\geq 3$ .

Possible Answers:

$(x-3)=\frac{x^2-9}{x+3}$

The rational expression is undefined.

None of the other answers are correct.

$(x-3)> \frac{x^2-9}{x+3}$

$(x-3)< \frac{x^2-9}{x+3}$

Correct answer:

$(x-3)=\frac{x^2-9}{x+3}$

Explanation:

$(x-3)\circ\frac{x^2-9}{x+3}$

Let us simplify the second expression. We know that:

$(x^2-9)=(x+3)(x-3)$

So we can cancel out as follows:

$\frac{x^2-9}{x+3}=\frac{(x+3)(x-3)}{(x+3)}=x-3$

$(x-3)=\frac{x^2-9}{x+3}$

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Example Question #1 : Fractions

What value must take in order for the following expression to be greater than zero?

$\frac{3}{k}-\frac{5}{7}$

Possible Answers:

Correct answer:

Explanation:

is such that:

$\frac{3}{k}-\frac{5}{7}>0$

Add $\frac{5}{7}$ to each side of the inequality:

$\frac{3}{k}>\frac{5}{7}$

Multiply each side of the inequality by :

$\frac{21}{k}>5$

Multiply each side of the inequality by :

21>5k

Divide each side of the inequality by :

$k< \frac{21}{5}$

You can now change the fraction on the right side of the inequality to decimal form.

$k<\frac{21}{5}=\frac{20}{5}+\frac{1}{5}= 4.2$

The correct answer is , since k has to be less than 4.2 for the expression to be greater than zero.

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Example Question #155 : Algebra

Give the solution set of this inequality:

$|-4x+ 17| \ge 67$

Possible Answers:

$(-\infty, -21 ] \cup [12.5, \infty )$

The set of all real numbers

[-12.5, 21]

[-21, 12.5]

$(-\infty, -12.5 ] \cup [21, \infty )$

Correct answer:

$(-\infty, -12.5 ] \cup [21, \infty )$

Explanation:

The absolute value inequality

$|-4x+ 17| \ge 67$

can be rewritten as the compound inequality

$-4x+ 17 \le - 67$ or $-4x+ 17 \ge 67$

Solve each inequality separately, using the properties of inequality to isolate the variable on the left side:

$-4x+ 17 \le - 67$

Subtract 17 from both sides:

$-4x+ 17 - 17 \le - 67 - 17$

$-4x \le - 84$

Divide both sides by , switching the inequality symbol since you are dividing by a negative number:

$-4x \div (-4) \ge - 84 \div (-4)$

$x \ge 21$ ,

which in interval notation is $[21, \infty )$

The same steps are performed with the other inequality:

$-4x+ 17 \ge 67$

$-4x+ 17- 17 \ge 67 - 17$

$-4x \ge 50$

$-4x \div (-4) \le 50 \div (-4)$

$x \le -12.5$

which in interval notation is $(-\infty, -12.5 ]$ .

The correct response is the union of these two sets, which is

$(-\infty, -12.5 ] \cup [21, \infty )$ .

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Example Question #1931 : Sat Mathematics

Find the maximum value of f(x,y) , from the system of inequalities.

$y\geq 2$

$4x+5\geq y+3$

$1\leq x\leq5$

f(x,y)=3x+3y

Possible Answers:

Correct answer:

Explanation:

First step is to rewrite $4x+5\geq y+3$

$4x+5\geq y+3$

$4x+2\geq y$

Next step is to find the vertices of the bounded region. We do this by plugging in the x bounds into the equation. Don't forgot to set up the other x and y bounds, which are given pretty much.

The vertices are

(1,2)

(5,2)

(1, 7)

(5, 22)

Now we plug each coordinate into f(x,y) , and what the maximum value is.

(1,2)=3(1)+3(2)=9

(5,2)=3(5)+3(1)=21

(1, 7)=3(1)+3(7)=25

(5, 22)=3(5)+3(22)=81

So the maximum value is

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Example Question #1 : How To Find The Solution To An Inequality With Division

Each of the following is equivalent to

xy/z * (5(x + y)) EXCEPT:

Possible Answers:

5x²y + 5xy²/z

xy(5x + 5y)/z

5x² + y²/z

xy(5y + 5x)/z

Correct answer:

5x² + y²/z

Explanation:

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1. We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z. xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression. 5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out. Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

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Example Question #3 : Inequalities

Let S be the set of numbers that contains all of values of x such that 2x + 4 < 8. Let T contain all of the values of x such that -2x +3 < 8. What is the sum of all of the integer values that belong to the intersection of S and T?

Possible Answers:

-3

-7

-2

Correct answer:

-2

Explanation:

First, we need to find all of the values that are in the set S, and then we need to find the values in T. Once we do this, we must find the numbers in the intersection of S and T, which means we must find the values contained in BOTH sets S and T.

S contains all of the values of x such that 2x + 4 < 8. We need to solve this inequality.

2x + 4 < 8

Subtract 4 from both sides.

2x < 4

Divide by 2.

x < 2

Thus, S contains all of the values of x that are less than (but not equal to) 2.

Now, we need to do the same thing to find the values contained in T.

-2x + 3 < 8

Subtract 3 from both sides.

-2x < 5

Divide both sides by -2. Remember, when multiplying or dividing an inequality by a negative number, we must switch the sign.

x > -5/2

Therefore, T contains all of the values of x that are greater than -5/2, or -2.5.

Next, we must find the values that are contained in both S and T. In order to be in both sets, these numbers must be less than 2, but also greater than -2.5. Thus, the intersection of S and T consists of all numbers between -2.5 and 2.

The question asks us to find the sum of the integers in the intersection of S and T. This means we must find all of the integers between -2.5 and 2.

The integers between -2.5 and 2 are the following: -2, -1, 0, and 1. We cannot include 2, because the values in S are LESS than but not equal to 2.

Lastly, we add up the values -2, -1, 0, and 1. The sum of these is -2.

The answer is -2.

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Example Question #212 : Gre Quantitative Reasoning

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Possible Answers:

$\dpi{100} \small x<35$

$\dpi{100} \small x<27$

$\dpi{100} \small x>27$

$\dpi{100} \small x<9$

$\dpi{100} \small x>9$

Correct answer:

$\dpi{100} \small x<9$

Explanation:

We simplify this inequality similarly to how we would simplify an equation

$\dpi{100} \small 3x+8-8<35-8$

$\dpi{100} \small \frac{3x}{3}<\frac{27}{3}$

Thus $\dpi{100} \small x<9$

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Example Question #6 : Inequalities

What is a solution set of the inequality 2x+12>42 ?

Possible Answers:

x>4

x>15

x<9

$x>\frac{3}{2}$

x<15

Correct answer:

x>15

Explanation:

In order to find the solution set, we solve 2x+12>42 as we would an equation:

2x+12>42

2x>30

x>15

Therefore, the solution set is any value of x>15 .

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Example Question #13 : Solving Inequalities

Which of the following could be a value of , given the following inequality?

$8-7x\leq11x+4$

Possible Answers:

$-\frac{2}{9}$

$\frac{1}{9}$

$\frac{4}{9}$

$-\frac{1}{3}$

$-\frac{2}{3}$

Correct answer:

$\frac{4}{9}$

Explanation:

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

Start by moving your variables to one side of the inequality and all other numbers to the other side:

$-7x-11x\leq4-8$

$-18x\leq-4$

Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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Example Question #1932 : Sat Mathematics

If $2\leq n \leq12$ and $9\leq p \leq10$ , which of the following gives the set of possible values of $\frac{n}{p}$ ?

Possible Answers:

$\frac{1}{5} \leq \frac{n}{p} \leq \frac{6}{5}$

$\frac{1}{5}\leq \frac{n}{p} \leq \frac{4}{3}$

$\frac{2}{9} \leq \frac{n}{p} \leq \frac{5}{6}$

$0 \leq \frac{n}{p} \leq \frac{1}{5}$

$\frac{2}{9} \leq \frac{n}{p} \leq \frac{6}{5}$

Correct answer:

$\frac{1}{5}\leq \frac{n}{p} \leq \frac{4}{3}$

Explanation:

To get the lowest value, you need the lowest numerator and the highest denominator. That would be $\frac{2}{10}$ or reduced to be $\frac{1}{5}$ . For the highest value, you need the highest numerator and the lowest denominator. That would be $\frac{12}{9}$ or $\frac{4}{3}$ .

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SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #6 : How To Find The Solution To An Inequality With Multiplication

Example Question #1 : Fractions

Example Question #155 : Algebra

Example Question #1931 : Sat Mathematics

Example Question #1 : How To Find The Solution To An Inequality With Division

Example Question #3 : Inequalities

Example Question #212 : Gre Quantitative Reasoning

Example Question #6 : Inequalities

Example Question #13 : Solving Inequalities

Example Question #1932 : Sat Mathematics

All SAT Math Resources

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