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GRE Subject Test: Math : Inequalities

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Example Question #1 :Absolute Value Inequalities

The weight of the bowling balls manufactured at the factory must belbs. with a tolerance oflbs. Which of the following absolute value inequalities can be used to assess which bowling balls are tolerable?

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #4 :Absolute Value Inequalities

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

Possible Answers:

x> 3.2 and x< -2.3

x<3.2 and x> -2.3

There is no solution.

x<-3.2 and x> -2.3

Correct answer:

x<3.2 and x> -2.3

Explanation:

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

4x + 1.2 - 3 < 11

4x -1.8<11

4x -1.8 +1.8<11 + 1.8

4x< 12.8

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

x<3.2

4x + 1.2 - 3> -11

4x - 1.8> -11

4x - 1.8 + 1.8> -11 + 1.8

4x > -9.2

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

x> -2.3

The correct answer is x<3.2 and x>2.3

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Example Question #5 :Absolute Value Inequalities

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

Possible Answers:

x<-2 and x> -10

x>-2 and x< -10

x<2 and x>10

x< 2 and x > 10

Correct answer:

x<-2 and x> -10

Explanation:

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

6 + x -4 < 0

6 + x < 4

6-6 + x < 4-6

x< -2

6 + x > -4

6-6 + x > -6-4

x>-10

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

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Example Question #6 :Absolute Value Inequalities

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

Possible Answers:

There is no solution.

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

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

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

Correct answer:

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

Explanation:

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

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

$x-6 \leq 6$

$x\leq 12$

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

$x-6 \geq -6$

$x\geq 0$

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

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Example Question #7 :Absolute Value Inequalities

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

Possible Answers:

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

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

x<12 and x> -2

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

Correct answer:

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

Explanation:

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

$x-5 \leq 7$

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

$x\leq 12$

$x-5 \geq -7$

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

$x\geq -2$

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

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Example Question #8 :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:

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

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

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

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

Correct answer:

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

Explanation:

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

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

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Example Question #9 :Absolute Value Inequalities

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

Possible Answers:

$x \le 6$

x>6

$x \ge 6$

x < 6

Correct answer:

$x \le 6$

Explanation:

Step 1: Separate the equation into two equations:

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

Step 2: Solve the first equation

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

Step 3: Solve the second equation

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

$-x \ge -6$

$\rightarrow x \le 6$

The solution is $x\le 6$

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Example Question #10 :Absolute Value Inequalities

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

Possible Answers:

x>19 and x< -5

1<x<19

-5<x<19

x<1 and x>19

Correct answer:

1<x<19

Explanation:

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

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

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

x-10<9 means that:

x<19

For the second:

x-10>-9 means that:

x>1

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

1<x<19

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Example Question #11 :Absolute Value Inequalities

$\frac{2}{3}\left | 9x+6\right | -2\geq 1$

Possible Answers:

$x\leq-\frac{1}{6}$ and $x\geq- \frac{1}{2}$

$x\geq \frac{1}{6}$ and $x\leq \frac{1}{2}$

$x\geq -\frac{1}{6}$ or $x\leq -\frac{7}{6}$

There is no solution.

Correct answer:

$x\geq -\frac{1}{6}$ or $x\leq -\frac{7}{6}$

Explanation:

$\frac{2}{3}\left | 9x+6\right | -2\geq 1$

$\frac{2}{3}\left | 9x+6 \right |\geq 3$

$\left | 9x+6 \right |\geq \frac{9}{2}$

At this point, you've isolated the absolute value and can solve this problems for both cases, $9x+6\geq \frac{9}{2}$ and $9x+6\leq -\frac{9}{2}$ . Beginning with the first case:

$9x+6\geq \frac{9}{2}$

$9x+\frac{12}{2}\geq\frac{9}{2}$

$9x\geq -\frac{3}{2}$

$x\geq -\frac{1}{6}$

Then for the second case:

$9x+6\leq -\frac{9}{2}$

$9x+\frac{12}{2}\leq -\frac{9}{2}$

$9x\leq -\frac{21}{2}$

$x\leq -\frac{7}{6}$

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Example Question #12 :Absolute Value Inequalities

$\left | -4x-5 \right |< 11$

Possible Answers:

x>-11 or x<-6

x> 4

x< -4

$x< \frac{4}{5}$ or x>-2

x> -4 and $x< \frac{3}{2}$

Correct answer:

x> -4 and $x< \frac{3}{2}$

Explanation:

自从x的绝对值是独自one side of the inequality, you set the expression inside the absolute value equal to both the positive and negative value of the other side, 11 and -11 in this case. For the negative value -11, you must also flip the inequality from less than to a greater than. You should have two inequalities looking like this.

-4x-5<11 and -4x-5>-11

Add 5 to both sides in each inequality.

-4x<16 and -4x>-6

Divide by -4 to both sides of the inequality. Remember, dividing by a negative will flip both inequality symbols and you should have this.

x>-4 and $x< \frac{3}{2}$

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GRE Subject Test: Math : Inequalities

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 #4 :Absolute Value Inequalities

Example Question #5 :Absolute Value Inequalities

Example Question #6 :Absolute Value Inequalities

Example Question #7 :Absolute Value Inequalities

Example Question #8 :Absolute Value Inequalities

Example Question #9 :Absolute Value Inequalities

Example Question #10 :Absolute Value Inequalities

Example Question #11 :Absolute Value Inequalities

Example Question #12 :Absolute Value Inequalities

All GRE Subject Test: Math Resources

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