Absolute value

Absolute Value Expressions I

Submitted by ladislav.foltyn on Fri, 02/15/2019 - 15:00

Question:

\footnotesize Evaluate absolute value expressions:

1003187106

Level:

Consider expressions

| 3 x - 12 |

3 | x | + 12

| 3 x | - | - 12 |

3 | x - 4 |

. Which of the expressions has the greatest value if we substitute any

x

from the interval

(0; + \infty)

3 | x | + 12

| 3 x - 12 |

| 3 x | - | - 12 |

3 | x - 4 |

1003187105

Level:

Let

a

be a positive real number. Then

| x | \geq a

if and only if

x \geq a

x \leq - a

if and only if

x \geq a

if and only if

x \leq - a

if and only if

x > 0

1003187104

Level:

Let

a

be a positive real number. Then

| x | \leq a

if and only if

- a \leq x \leq a

if and only if

x \leq a

if and only if

x \geq - a

if and only if

x < 0

1003187103

Level:

Find the relation that does not hold for any

x

y \in R

| | x | - | y | | > | x + y |

| x y | = | x | | y |

| \frac{x}{y} | = \frac{| x |}{| y |}, y \neq 0 .

| (x y)^{2} | = | x y |^{2} = (x y)^{2}

1003187102

Level:

For

x

y \in R

consider

| x + y | = | x | + | y |

The equality holds if and only if the sign of

x

and

y

is the same.

The equality does not hold for any

x

and

y

The equality holds if and only if

x

and

y

are all positive.

The equality holds if and only if

x

and

y

are both nonpositive.

1003187101

Level:

Which of the following relations is correct for all

x

y \in R

| x + y | \leq | x | + | y |

| x + y | = | x | + | y |

| x - y | < | x | - | y |

| x - y | = | x | - | y |

1003187304

Level:

How many solutions does the equation

| | x - 4 | - 2 | + 2 = 0

have?

0

2

4

6

1003187405

Level:

Choose the expression which takes only negative values.

- | 2 - 4 x | - 1

| - 2 x - 5 | - 2

- | 2 - 4 x | + 2

| 2 - x | - 2

1003187004

Level:

Let

x \in (- \infty; 0)

. The expression

| x - | x | | + | x + | x | | + 1 - x | x |

is equal to:

(x - 1)^{2}

(x + 1)^{2}

x^{2} + 1

1 - x^{2}

Absolute Value Expressions I

1003187106

1003187105

1003187104

1003187103

1003187102

1003187101

1003187304

1003187405

1003187004

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