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GRE Subject Test: Math : Classifying Algebraic Functions

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Example Questions

Example Question #1 : Absolute Value Inequalities

$5\left | x-7\right |+14<39$ $\displaystyle 5\left | x-7\right |+14< 39$

Possible Answers:

x<12 $\displaystyle x< 12$

2< x<12 $\displaystyle 2< x< 12$

x> 12 $\displaystyle x> 12$ or x< 2 $\displaystyle x< 2$

2< x $\displaystyle 2< x$

Correct answer:

2< x<12 $\displaystyle 2< x< 12$

Explanation:

The first thing we must do is get the absolute value alone: $5\left | x-7\right |+14<39$ $\displaystyle 5\left | x-7\right |+14< 39$

$5\left | x-7\right |<25$ $\displaystyle 5\left | x-7\right |< 25$

$\left | x-7\right |<5$ $\displaystyle \left | x-7\right |< 5$

When we're working with absolute values, we are actually solving two equations:

x-7<5 $\displaystyle x-7< 5$ and x-7> -5 $\displaystyle x-7> -5$

Fortunately, these can be written as one equation:

-5< x-7<5 $\displaystyle -5< x-7< 5$

If you feel more comfortable solving the equations separately then go ahead and do so.

2< x<12 $\displaystyle 2< x< 12$

To get $\displaystyle x$ alone, we added $\displaystyle 7$ on both sides of the inequality sign

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Example Question #2 : Absolute Value Inequalities

$\left | y\right | + 18 \leq 11.3$ $\displaystyle \left | y\right | + 18 \leq 11.3$

Possible Answers:

There is no solution.

$y\geq 6.7$ $\displaystyle y\geq 6.7$

$y\leq 29.3$ $\displaystyle y\leq 29.3$

$y\leq -6.7$ $\displaystyle y\leq -6.7$

Correct answer:

There is no solution.

Explanation:

$\left | y\right | + 18 \leq 11.3$ $\displaystyle \left | y\right | + 18 \leq 11.3$

$\left | y\right |\leq -6.7$ $\displaystyle \left | y\right |\leq -6.7$

Because Absolute Value must be a non-negative number, there is no solution to this Absolute Value inequality.

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Example Question #1 : Absolute Value Inequalities

The weight of the bowling balls manufactured at the factory must be $\displaystyle 8$ lbs. with a tolerance of $\displaystyle 3$ lbs. Which of the following absolute value inequalities can be used to assess which bowling balls are tolerable?

Possible Answers:

$\left | w-8\right | \leq 3$ $\displaystyle \left | w-8\right | \leq 3$

$\left | w-8\right | \geq 3$ $\displaystyle \left | w-8\right | \geq 3$

$\left | w-3 \right | \leq 8$ $\displaystyle \left | w-3 \right | \leq 8$

$\left | w-3 \right |\geq 8$ $\displaystyle \left | w-3 \right |\geq 8$

Correct answer:

$\left | w-8\right | \geq 3$ $\displaystyle \left | w-8\right | \geq 3$

Explanation:

The following absolute value inequality can be used to assess the bowling balls that are tolerable:

$\left | w-8\right |\geq3$ $\displaystyle \left | w-8\right |\geq3$

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Example Question #31 : Inequalities

$4\left |(x+0.3)\right | - 3 < 11$ $\displaystyle 4\left |(x+0.3)\right | - 3 < 11$

Possible Answers:

There is no solution.

x<3.2 $\displaystyle x< 3.2$ and x> -2.3 $\displaystyle x> -2.3$

x> 3.2 $\displaystyle x> 3.2$ and x< -2.3 $\displaystyle x< -2.3$

x<-3.2 $\displaystyle x< -3.2$ and x> -2.3 $\displaystyle x> -2.3$

Correct answer:

x<3.2 $\displaystyle x< 3.2$ and x> -2.3 $\displaystyle x> -2.3$

Explanation:

$4\left |(x+0.3) \right | - 3< 11$ $\displaystyle 4\left |(x+0.3) \right | - 3< 11$

$\displaystyle 4x + 1.2 - 3 < 11$

4x -1.8<11 $\displaystyle 4x -1.8< 11$

4x -1.8 +1.8<11 + 1.8 $\displaystyle 4x -1.8 +1.8< 11 + 1.8$

$\displaystyle 4x< 12.8$

$\frac{4x}{4} < \frac{12.8}{4}$ $\displaystyle \frac{4x}{4} < \frac{12.8}{4}$

x<3.2 $\displaystyle x< 3.2$

$\displaystyle 4x + 1.2 - 3> -11$

$\displaystyle 4x - 1.8> -11$

$\displaystyle 4x - 1.8 + 1.8> -11 + 1.8$

$\displaystyle 4x > -9.2$

$\frac{4x}{4}>\frac{-9.2}{4}$ $\displaystyle \frac{4x}{4}>\frac{-9.2}{4}$

x> -2.3 $\displaystyle x> -2.3$

The correct answer is x<3.2 $\displaystyle x< 3.2$ and x>2.3 $\displaystyle x>2.3$

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Example Question #2 : Absolute Value Inequalities

$\left | 6+x\right |-4 < 0$ $\displaystyle \left | 6+x\right |-4 < 0$

Possible Answers:

x<2 $\displaystyle x< 2$ and x>10 $\displaystyle x>10$

x>-2 $\displaystyle x>-2$ and x< -10 $\displaystyle x< -10$

x<-2 $\displaystyle x< -2$ and x> -10 $\displaystyle x> -10$

x< 2 $\displaystyle x< 2$ and x > 10 $\displaystyle x > 10$

Correct answer:

x<-2 $\displaystyle x< -2$ and x> -10 $\displaystyle x> -10$

Explanation:

$\left | 6+x\right | -4 < 0$ $\displaystyle \left | 6+x\right | -4 < 0$

$\displaystyle 6 + x -4 < 0$

$\displaystyle 6 + x < 4$

$\displaystyle 6-6 + x < 4-6$

x< -2 $\displaystyle x< -2$

$\displaystyle 6 + x > -4$

$\displaystyle 6-6 + x > -6-4$

x>-10 $\displaystyle x>-10$

The correct answer is x< -2 $\displaystyle x< -2$ and x>-10. $\displaystyle x>-10.$

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Example Question #1 : Absolute Value Inequalities

$\left | \frac{x-6}{2} \right |\leq 3$ $\displaystyle \left | \frac{x-6}{2} \right |\leq 3$

Possible Answers:

$x\geq 0$ $\displaystyle x\geq 0$ and $x\leq 12$ $\displaystyle x\leq 12$

There is no solution.

$x\geq 0$ $\displaystyle x\geq 0$ and $x\leq 12$ $\displaystyle x\leq 12$

$x\leq 0$ $\displaystyle x\leq 0$ and $x\geq12$ $\displaystyle x\geq12$

Correct answer:

$x\geq 0$ $\displaystyle x\geq 0$ and $x\leq 12$ $\displaystyle x\leq 12$

Explanation:

$\left | \frac{x-6}{2} \right |\leq3$ $\displaystyle \left | \frac{x-6}{2} \right |\leq3$

$\frac{2}{1} \left | \frac{x-6}{2} \right | \leq 2\times3$ $\displaystyle \frac{2}{1} \left | \frac{x-6}{2} \right | \leq 2\times3$

$x-6 \leq 6$ $\displaystyle x-6 \leq 6$

$x\leq 12$ $\displaystyle x\leq 12$

$\frac{2}{1} \left | \frac{x-6}{2} \right | \geq 2 \times -3$ $\displaystyle \frac{2}{1} \left | \frac{x-6}{2} \right | \geq 2 \times -3$

$x-6 \geq -6$ $\displaystyle x-6 \geq -6$

$x\geq 0$ $\displaystyle x\geq 0$

The correct answer is $x\leq12$ $\displaystyle x\leq12$ and $x\geq 0.$ $\displaystyle x\geq 0.$

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Example Question #31 : Inequalities

$\left | x-5\right |\leq7$ $\displaystyle \left | x-5\right |\leq7$

Possible Answers:

$x\geq12$ $\displaystyle x\geq12$ and $x\leq -2$ $\displaystyle x\leq -2$

$x\leq-12$ $\displaystyle x\leq-12$ and $x \geq 2$ $\displaystyle x \geq 2$

x<12 $\displaystyle x< 12$ and x> -2 $\displaystyle x> -2$

$x\leq12$ $\displaystyle x\leq12$ and $x\geq-2$ $\displaystyle x\geq-2$

Correct answer:

$x\leq12$ $\displaystyle x\leq12$ and $x\geq-2$ $\displaystyle x\geq-2$

Explanation:

$\left | x-5\right |\leq 7$ $\displaystyle \left | x-5\right |\leq 7$

$x-5 \leq 7$ $\displaystyle x-5 \leq 7$

$x-5 + 5 \leq 7 +5$ $\displaystyle x-5 + 5 \leq 7 +5$

$x\leq 12$ $\displaystyle x\leq 12$

$x-5 \geq -7$ $\displaystyle x-5 \geq -7$

$x-5 +5 \geq -7 +5$ $\displaystyle x-5 +5 \geq -7 +5$

$x\geq -2$ $\displaystyle x\geq -2$

The correct answer is $x\leq12$ $\displaystyle x\leq12$ and $x\geq -2.$ $\displaystyle x\geq -2.$

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Example Question #2 : Absolute Value Inequalities

A type of cell phone must be less than 9 ounces with a tolerance of 0.4 ounces. Which of the following inequalities can be used to assess which cell phones are tolerable? (w refers to the weight).

Possible Answers:

$\left | w-9\right | \leq 0.4$ $\displaystyle \left | w-9\right | \leq 0.4$

$\left | w-0.4\right | \leq 9$ $\displaystyle \left | w-0.4\right | \leq 9$

$\left | w-0.4\right | \geq 9$ $\displaystyle \left | w-0.4\right | \geq 9$

$\left | w-9\right | \geq 0.4$ $\displaystyle \left | w-9\right | \geq 0.4$

Correct answer:

$\left | w-9\right | \leq 0.4$ $\displaystyle \left | w-9\right | \leq 0.4$

Explanation:

The Absolute Value Inequality that can assess which cell phones are tolerable is:

$\left | w-0.4\right | \leq 9$ $\displaystyle \left | w-0.4\right | \leq 9$

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Example Question #2 : Absolute Value Inequalities

Solve for x: $|x-1|\le 5$ $\displaystyle |x-1|\le 5$

Possible Answers:

$x \le 6$ $\displaystyle x \le 6$

$x \ge 6$ $\displaystyle x \ge 6$

x < 6 $\displaystyle x < 6$

x>6 $\displaystyle x>6$

Correct answer:

$x \le 6$ $\displaystyle x \le 6$

Explanation:

Step 1: Separate the equation into two equations:

First Equation: $|x-1|\le 6$ $\displaystyle |x-1|\le 6$
Second Equation: $-(x-1)\ge -5$ $\displaystyle -(x-1)\ge -5$

Step 2: Solve the first equation

$x-1\le 5$ $\displaystyle x-1\le 5$
$x-1+1\le 5+1$ $\displaystyle x-1+1\le 5+1$
$x \le 6$ $\displaystyle x \le 6$

Step 3: Solve the second equation

$-(x+1)\ge -5$ $\displaystyle -(x+1)\ge -5$
$-x+1\ge -5$ $\displaystyle -x+1\ge -5$
$-x+1-1\ge -5-1$ $\displaystyle -x+1-1\ge -5-1$

$-x \ge -6$ $\displaystyle -x \ge -6$

$\rightarrow x \le 6$ $\displaystyle \rightarrow x \le 6$

The solution is $x\le 6$ $\displaystyle x\le 6$

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Example Question #1 : Absolute Value Inequalities

Which of the following expresses the entire solution set of $\left | x-10\right |+ 3 < 12$ $\displaystyle \left | x-10\right |+ 3 < 12$?

Possible Answers:

x<1 $\displaystyle x< 1$ and x>19 $\displaystyle x>19$

x>19 $\displaystyle x>19$ and x< -5 $\displaystyle x< -5$

1<x<19 $\displaystyle 1< x< 19$

-5<x<19 $\displaystyle -5< x< 19$

Correct answer:

1<x<19 $\displaystyle 1< x< 19$

Explanation:

Before expanding the quantity within absolute value brackets, it is best to simplify the "actual values" in the problem. Thus $\left | x-10 \right |+3<12$ $\displaystyle \left | x-10 \right |+3< 12$ becomes:

$\left | x-10 \right |<9$ $\displaystyle \left | x-10 \right |< 9$

From there, note that the absolute value means that one of two things is true: x-10<9 $\displaystyle x-10< 9$ or x-10>-9 $\displaystyle x-10>-9$. You can therefore solve for each possibility to get all possible solutions. Beginning with the first:

x-10<9 $\displaystyle x-10< 9$ means that:

x<19 $\displaystyle x< 19$

For the second:

x-10>-9 $\displaystyle x-10>-9$ means that:

x>1 $\displaystyle x>1$

Note that the two solutions can be connected by putting the inequality signs in the same order:

1<x<19 $\displaystyle 1< x< 19$

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GRE Subject Test: Math : Classifying Algebraic Functions

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Absolute Value Inequalities

Example Question #2 : Absolute Value Inequalities

Example Question #1 : Absolute Value Inequalities

Example Question #31 : Inequalities

Example Question #2 : Absolute Value Inequalities

Example Question #1 : Absolute Value Inequalities

Example Question #31 : Inequalities

Example Question #2 : Absolute Value Inequalities

Example Question #2 : Absolute Value Inequalities

Example Question #1 : Absolute Value Inequalities

All GRE Subject Test: Math Resources

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