Signed Numbers, Absolute Value, Coordinate Geometry

Question 1

What is a number line?

Answer 1

A number line is a line segment on which integers are listed in numerical order.

*** Integers include all positive and negative whole numbers, as well as zero.

Question 2

Define:

A signed number.

Answer 2

A signed number is a number with a plus or minus sign associated with it.

*** Special rules are needed for working with them.

*** When the sign is omitted, it is understood to be plus.

Question 3

Define:

The absolute value of a number.

Answer 3

The absolute value of a number is the distance between that number and zero on the number line.

|a| = a

if a is greater than or equal to zero.

|a| = -a

if a is less than zero.

Question 4

Is the absolute value of a sum of two numbers always equal to the sum of the absolute values?

|a + b| ? |a| + |b|

Answer 4

No, not always. The absolute value of a sum of two numbers is less than or equal to the sum of the absolute values.

|a + b| ≤ |a| + |b|

• Example:*
• I5 + 3I = I5I + I3I

I8I = 8 ; I5I + I3I = 5 + 3 = 8

• I5 + (-3)I I5I + I-3I

I2I = 2 ; I5I + I-3I = 5 + 3 = 8

Question 5

Is the absolute value of a product of two numbers always equal to the product of the absolute values?

|a x b| ? |a| x |b|

Answer 5

Yes, the absolute value of a product of two numbers always equals to the product of the absolute values.

|a x b| = |a| x |b|

Example:

I5 x (-3)I = I-15I = 15

I5I x I-3I = 5 x 3 = 15

Question 6

Fill in the blank:

Opposite numbers have …….. absolute values.

IaI ? I-aI

Answer 6

Opposite numbers have equal absolute values.

IaI = I-aI

Example:

I3I = I-3I = 3

Question 7

Define:

The distance between two real numbers on the number line.

d (a,b)

Answer 7

The distance between two real numbers a and b on the number line is defined as the absolute value of b - a.

• d (a,b) =* Ib - aI
• Example:*

d (5, -3) = I-3 - 5I = 8

Question 8

Fill in the blank:

The absolute value is always greater than or equal….?

Answer 8

The absolute value of a number is always greater than or equal zero.

Question 9

The sum of two positive numbers is positive. What sign is the sum of two negative numbers?

• 5 + ( -3) = ?

Answer 9

When you add two negative numbers, add their absolute values. The sign of the sum is negative.

• Example:*
• 5 + (-3) = -8

|-5| + |-3| = 5 + 3 = 8

Question 10

How do you add two numbers with different signs?

Answer 10

When you add two numbers with different signs, subtract the absolute values (the smaller from the larger) and keep the sign of the number with the greater absolute value.

• Example:*
• 5 + 3 = -2

|-5| - |3| = 5 - 3 = 2

Since -5 has the greater absolute value than 3, the result is negative.

Question 11

What number is the negative of a negative number?

-(-a) = ?

Answer 11

The negative of a negative number is the opposite positive number.

• (-a) = + a

Question 12

How do you subtract two numbers with the same sign?

3 - 7 = ?

Answer 12

To subtract two numbers with the same sign, rewrite the problem as follows:

(a - b) = a + (-b)

Follow the rules for adding signed numbers.

Example:

3 - 7 = 3 + (-7) = -4

I-7I - I3I = 7 - 3 = 4

*** The result is negative since the negative number in this case has a greater absolute value.

Question 13

How do you subtract two numbers with different signs?

3 - (-7) = ?

Answer 13

Subtract an integer by adding its opposite. The negative of a negative number is the opposite positive number.

• (-7) = +7

3 + 7 = 10

Question 14

How do you multiply or divide two numbers with the same sign?

What sign is the product or the quotient of two same sign numbers?

Answer 14

• Find the product or the quotient of the absolute values of the two numbers
• Two numbers with the same sign always give you a positive result

Example:

5 x 3 = 15

(-5) x (-3) = 15

|5| = |-5| = 5 ; |3| = |-3| = 3

Question 15

How do you multiply or divide two numbers with different signs?

What sign is the product or the quotient of two numbers with different signs?

Answer 15

• Find the product or the quotient of the absolute values of the two numbers
• Two numbers with different signs always produce a negative result

Example:

(-5) x 3 = -15

|-5| = 5 ; |3| = 3

Question 16

How do you add more than two signed numbers?

-4 + 6 + 5 + (-8) = ?

Answer 16

• 4 + 6 + 5 + (-8) = -1
• Work from left to right and add numbers two at a time, following the rules

-4 + 6 = 2
2 + 5 = 7
7 + (-8) = -1 or

• Add all positive numbers, then add all negative numbers, following the rules. Find the sum of your answers

6 + 5 = 11
- 4 + (-8) = -12
11 + (-12) = -1

Question 17

When you multiply more than two signed numbers, when is their product positive?

Answer 17

When you multiply signed numbers, find the product of the absolute values of all factors.

The product is positive when:

• all factors are positive
• there is an even number of negative factors.

Example:

(-5) x (2) x (-1) x (2) = 20

Question 18

When you multiply more than two signed numbers, when is the product negative?

Answer 18

Find the product of the absolute values of all factors.

The product is negative when there is an odd number of negative factors.

Example:

(-5) x (-2) x (-1) x (2) = -20

Question 19

What is the definition of a coordinate plane?

Answer 19

A coordinate plane is a plane formed by the intersection of a vertical (y-axis) and a horizontal (x-axis) number lines.

Question 20

What is a coordinate plane used for?

Answer 20

A coordinate plane is used to graph ordered pairs, straight lines and functions.

Question 21

What is the definition of an ordered pair?

Answer 21

An ordered pair is two numbers written in a specific order.

It is used to locate a point on a coordinate plane.

Usually denoted (x,y) where x and y are the coordinates of the point.

Example:

A (3, -8)

B (-6, -2)

Question 22

How do you plot a point on a coordinate plane?

Answer 22

To plot a point, you need to know the x and y coordinates of that point (the ordered pair).

• a point with (0,0) coordinates is the origin
• a point with (0,y) is the point on y-axis
• a point with (x,0) is the point on x-axis

Question 23

There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.

In which quadrant are both x and y coordinates of a point

• positive?
• negative?

Answer 23

• The x and y coordinates of a point are both positive in the first quadrant (QI)
• The x and y coordinates of a point are both negative in the third quadrant (QIII)
• x* > 0; y > 0 - first quadrant
• x* y

Question 24

There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.

In which quadrant(s) do the x and y coordinates of a point have different signs?

Answer 24

The x and y coordinates of a point have different signs in the second and the fourth quadrants (Q II and Q IV).

• x* y > 0 - second quadrant
• x* > 0; y

Question 25

What type of graph represents a **linear equation**?

Answer 25

A **linear equation** is an equation whose graph is a straight line.

Question 26

Write a standard form of a linear equation.

Answer 26

***ax* + *by* = *c*** ## Footnote where *a, b* and *c* are integers and both *a* and *b* *cannot* equal zero.

Question 27

What is *y*-intercept?

Answer 27

*y*-intercept of a line is the point where this line crosses the *y*-axis.

Question 28

What equation represents **slope-intercept** form of a linear equation?

Answer 28

***y* = *mx* + *b*** ## Footnote * m* stands for the **slope,** * b* is ***y*-intercept.**

Question 29

How can you prove that *b* is the y-intercept of a line in the equation below? ***y = mx + b***

Answer 29

By definitions, *y*-intercept of a line is the point where this line crosses *y*-axis. Therefore, the *x* coordinate of that point is zero. Plug in zero in the equation below and you'll see that *y* equals *b*. ***y* = *mx* + *b*** *y = m \** 0 *+ b ⇒ y = b*

Question 30

# Define: The slope of a line.

Answer 30

The slope (*m*) of a line measures the steepness of that line. * y = **m**x + b* * m* is a ratio that shows the change in value of *y* to the change of value of *x*. * m* = ![]()

Question 31

When is the slope of a line: * negative? * positive?

Answer 31

* When the slope of a line is *negative*, the line goes *down* left to right. * When the slope is *positive*, the line *rises up* left to right.

Question 32

# Fill in the blanks: If two lines have the same graph, * their _____ are \_\_\_\_\_\_ and * their _____ are ______ .

Answer 32

If two lines have the same graph, their *slopes* are *equal* and their *y-intercepts* are *equal*.

Question 33

When two lines are parallel, their slopes are....?

Answer 33

Parallel lines have *equal* slopes.

Question 34

When two lines are perpendicular to each other, their slopes are....?

Answer 34

Perpendicular lines have the slopes that are *opposite* *reciprocals* of each other. The product of the slopes of perpendicular lines equals -1. m₁ x m₂ = -1 \*\*\* m₁ and m₂ - slopes of l₁ and l₂ l₁ and l₂ are perpendicular, nonvertical, nonhorizontal straight lines.

Question 35

What is the equation of a vertical line? What is the equation of a horizontal line?

Answer 35

* Vertical line: ***x* = k** * Horizontal line: ***y* = k** \*\*\* k is any real number * Example:* * y* = 2. This line runs parallel to the x-axis and for all its *x*-coordinates the *y*-coordinate is 2.

Question 36

What is the slope of a vertical line?

Answer 36

Vertical lines have *no* slope or an *undefined* slope.

Question 37

What is the slope of a horizontal line?

Answer 37

Horizontal lines have the slope of *zero.*

Question 38

# Write: The *point-slope* form of a linear equation.

Answer 38

This is the **point-slope** form of a linear equation: * y - y₁* = *m (x - x₁)* * m* - the slope of a line * (x₁, y₁)* - point on the line

Question 39

How do you determine the coordinates of the point *halfway* between two points?

Answer 39

Use the **midpoint formula**:

where (*x₁; y₁*) and (*x₂; y₂*) are the coordinates of the end points. \*\*\*This concept will help you remember the formula. Think of it as finding the *average* of the *x* and *y*-coordinates of the end points.

Question 40

How do you determine the distance between two points in a coordinate graph?

Answer 40

Use the **distance formula**: ## Footnote

*d* - distance between two points with (*x₁;y₁*) and (*x₂;y₂*) coordinates. \*\*\* This concept will help you remember the distance formula: *It is a variant of the Pythagorean Theorem***.** You can create a right triangle connecting two points to find the hypotenuse. The hypotenuse will be the distance between two points.

Question 41

In which quadrant(s) lie the points with the coordinates *x* and *y* such as *xy*

Answer 41

Those points lie in the second and the fourth quadrants (Q II and Q IV). ## Footnote The question is asking you to find the points with different sign coordinates since the product of *x* and *y* must be negative. * x* y \> 0 - second quadrant * x* \> 0; *y*

Question 42

How do you find the slope of a line when given the coordinates of two points on that line? point A (3;4) point B (2;1)

Answer 42

The slope of a line is the rise of that line over the run. The rise: *y_B - y_A* = 1 - 4 = -3 The run: *x_B - x_A *= 2 - 3 = -1 The slope is ^-3*/*_-1 = 3.

Question 43

How do you find the slope of a line when given the standard form of linear equation for that line? 3*x* + 6*y* = 12

Answer 43

Rewrite this equation in the *slope-intercept* form. In other words, isolate the *y* variable. *y = mx + b* 3*x* + 6*y* = 12 ⇒ 6*y* = 12 - 3*x* y = 2 - ¹*/*₂*x* = - ¹*/*₂*x* + 2 The slope is -¹*/*₂.

Question 44

Find the *y*-intercept of the line described by the equation below. 8*x* + 2*y* = 16

Answer 44

The *y*-intercept is the point with (0, 8) coordinates. * By definition, the *y*-intercept of a line has (0, *y*) coordinates. Plug in 0 into the equation and solve for *y*. 2*y* = 16 ⇒ *y* = 8 * Or you can rewrite the equation in the slope-intercept form. y = 4x + 8

Question 45

What can you tell about the line described by the equation below? *x* + *y* = -5

Answer 45

* This line crosses y-axis at a point (0, -5). * The slope of this line is -1. * The line has a negative slope, it falls from left to right. Rewrite the equation in the slope-intercept form. * y = mx + b* * x* + *y* = -5 ⇒ *y* = - *x - 5*

Question 46

Find the distance between points A and B with the following coordinates? A (-4; 3) and B (4; -3)

Answer 46

The distance between A and B equals 10. ## Footnote Use the distance formula to find the answer. ![]() (4-(-4))² + (-3-3)² = 64 + 36 = 100 Take square root of 100 and the result will be the distance between points A and B.

Question 47

What is the slope of a line parallel to the line with this equation? 3*x* + 6*y* = 18

Answer 47

The slope of the parallel line is -¹/₂. To find out the slope of a line described by the equation, rewrite the equation in the slope-intercept form: ***y = mx + b*** 6*y* = -3*x* + 18 ⇒ *y* = -¹*/*₂*x* + 3 Slopes of parallel lines are the *same*.

Question 48

What is the slope of a line perpendicular to the line described by the equation below? 6x + 2y = 12?

Answer 48

The slope of the perpendicular line is ¹/₃. Put the equation into the slope-intercept form: ``` ***y = mx + b*** 2*y* = -6*x* + 12 ⇒ *y* = -3*x* + 6 ``` The slopes of two perpendicular lines are *the opposite reciprocals* of one another. Since the slope of this line is -3, the slope of the perpendicular line is ¹*/*₃.

Question 49

Find solutions to the equation below. I2*x* - 5I = 7

Answer 49

*x* = 6; *x* = -1 ## Footnote The absolute value of 2*x* - 5 equals 7. It means that: 2*x* - 5 = 7 ⇒2*x* = 12 ⇒ *x* = 6 2*x* - 5 = -7 ⇒ 2*x* = -2 ⇒ *x* = -1

Question 50

What is the solution set of the inequality below? I2*x* - 5I \> 7

Answer 50

- 1 \> *x* \> 6 ## Footnote The absolute value of 2x - 5 is greater than 7. It means that: 2x - 5 \> 7 ⇒2x \> 12 ⇒ x \> 6 2x - 5 \> -7 ⇒ 2x \> -2 ⇒ x \*\*\* Remember, dividing by a negative number changes the direction of the inequality sign.

Question 51

# Solve: - (- 4 - 7) + (- 2) = ?

Answer 51

- (- 4 - 7) + (- 2) = - (-11) + (-2) = 11 - 2 = 9.

Question 52

* x, y* and *z* are distinct integers. * x* * y* * z = xy* Is *z* positive or negative?

Answer 52

Z is positive. Since *x and y x* and *y* must be negative integers. The product of two negative integers is a positive integer.

Question 53

If there are two positive numbers and three negative numbers in a multiplication sequence, what sign is the product of the five numbers?

Answer 53

The product is negative. ## Footnote positive x positive = positive negative x negative = positive positive x negative = negative *Example:* 2 x (-1) x 2 x (-3) x (-1) = -12

Question 54

Can you determine the sign of the result of the expression below? (-5) x 6 x (-4) x 0 x 8 = ?

Answer 54

No, you cannot. The result is zero and it doesn't have a sign. Zero is neither negative nor positive.

Question 55

Find the quotient and determine its sign. - ( - 132) / 6 = ?

Answer 55

22 The quotient is positive.

Question 56

Which point is *not* going to be on the line described by the following equation *y* = *kx* + *k*? ## Footnote (a) (1, 2*k*) (b) (0, *k*) (c) (2, 3*k*) (d) (0, *-k*)

Answer 56

(d) (0, *-k*) ## Footnote Realize that you are given the *x* and *y* coordinates of four points in question and the linear equation in the slope-intercept form. Plug in the coordinates of each point into the equation and find the point whose coordinates prove the equation false. For example, plugging in choice (c) you get 3k = k \* 2 + k; 3k = 3k which is true. Choice (d) results in -k = k which is false.

Question 57

Line *a* is perpendicular to line *b*. If the slope of line *a* is a positive integer, then the product of the slopes of lines *a* and *b* *cannot* equal ## Footnote (a) -5 (b) -1 (c) 1

Answer 57

(c) 1 ## Footnote Lines *a* and *b* are perpendicular which means that their slopes are the opposite reciprocals of each other. Therefore, their product *cannot* be a positive number.

Question 58

Line *m* is perpendicular to line *p*. If the slope of line *p* is a non-zero integer, then the slope of *p* divided by the slope of *m* *could* equal ## Footnote (a) ¹*/*₉ (b) 9 (c) -¹*/*₉ (d) -9

Answer 58

(d) -9 ## Footnote Lines *m* and *p* are perpendicular which means that their slopes are the opposite reciprocals of each other. Therefore, their quotient *cannot* be a positive number. Eliminate choices (a) and (b). Since the slope of line *p* is an integer, the slope of line *m* is a fraction. Dividing by a fraction is the same as multiplying by an integer. So, the right answer is an integer. The only choice that satisfies all conditions is choice (d).

Question 59

Find the sum of the two absolute values below: I*a*I + I*-a*I = ? (a) 2*a* (b) 2 I*a*I (c) I*a*I (d) I*-a*I

Answer 59

(b) 2 I*a*I The absolute value of *opposite* numbers is the *same*. Therefore, I*-a*I = I*a*I. I*a*I + I*a*I = 2 I*a*I\>

Question 60

There are four parallel lines. Each line has an integer slope. The product of their slopes *could* be...? ## Footnote (a) negative (b) prime (c) 0 (d) ¹*/*₈

Answer 60

The product of their slopes could be 0, choice (c). ## Footnote Parallel lines have *equal* slopes. The slopes of each of four lines can be positive or negative integers but their product will always be positive. Eliminate (a). The product of slopes cannot be a prime number since there are four factors. Eliminate (b). The product of four integers cannot be a fraction. Eliminate (d).

Question 61

If point A with coordinates (*x*, *y*) is in quadrant III, then point B with coordinates (*-x, -y*) is in quadrant ...? ## Footnote (a) Q I (b) Q II (c) Q III (d) Q IV

Answer 61

(a) Q I ## Footnote Since point A is located in Q III, both *x* and *y* are negative. Negative of a negative is positive therefore, the coordinates of point B are both positive. In Q I both *x* and *y* coordinates are positive.

Question 62

Any *x* which satisfies below conditions also satisfies this inequality. I2*x* - 5I - 7 \> 0 (a) *x* \> -6 (b) *x* (c) *x* \> 6 (d) *x*

Answer 62

(c) x \> 6 ## Footnote Isolate the absolute value expression and solve the inequality. I2x - 5I - 7 \> 0 ⇒ I2x - 5I \> 7 2x - 5 \> 7 ⇒2x \> 12 ⇒ x \> 6 2x - 5 \> -7 ⇒ 2x \> -2 ⇒ x *\*\*\* Remember, dividing by a negative number changes the direction of the inequality sign!!!*

Question 63

Write the equation of the line that is parallel to the line *y* = 36*x* - 12 and passes through the origin.

Answer 63

*y* = 36*x* If the line passes through the origin, its *y*-intercept is zero. Two lines are parallel, therefore, their slopes are equal.

Question 64

Determine the *x* and *y* coordinates of a point where the lines below cross. * y* = -12*x* - 3 * y* = 3*x* - 3 \*\*\* Don't graph the lines.

Answer 64

The lines cross at the point with (0, -3) coordinates. At the crossing point of two lines, their *x* and *y* are the same. * y* = -12*x* - 3 ; *y* = 3*x* - 3 - 12*x* - 3 = 3*x* - 3 This can only be true when *x* = 0. Plug in 0 into either equation to find *y*. *y* = -3.

Question 65

Without graphing, find the *x* and *y* intercepts of the line described by the following equation. *y* = -2*x* + 2

Answer 65

* The *x*-intercept of this line is 1 * The *y*-intercept of this line is 2 * y* equals zero at the *x*-intercept. - 2*x* + 2 = 0 ⇒ *x* = 1 * x* equals zero at the *y*-intercept. * y* = 2

Question 66

I2*x* - 6I = ## Footnote (a) 2*x* - 6 (b) 6 - 2*x* (c) I6 - 2*x*I (d) I2*x* + 6I

Answer 66

(c) I6 - 2*x*I ## Footnote I*a*I = I-*a*I, so I2*x* - 6I = I-(2*x* - 6)I = I-2*x* + 6I = I6 - 2*x*I

Question 67

The lines below have the same graph. *y* = 4*x* - 6 ¹*/*₂*y* = *mx + b* Find the product of *m* and *b.* *m \* b = ?*

Answer 67

*m \* b* = -6 The lines with the same graph have *equal* slopes and y-intercepts. You can either divide the first equation by 2 or multiply the second equation by 2 to compare them. * y* = 4*x* - 6 * y* = 2*mx* + 2*b* Since the slopes and the y-intercepts are equal, conclude that 2m equals 4 and 2b equals -6. * m* = 2; *b* = -3 * m \* b* = -6

Question 68

The product of 1,562 negative numbers *could* equal? ## Footnote (a) *k* (b) *-k* (c) 0 (d) -1

Answer 68

(a) *k* ## Footnote An even number of negative factors produces a positive product. Therefore, the product of 1,562 negative numbers must be positive. Eliminate (b), (c) and (d) choices.

Question 69

If the product of 1,563 real numbers is negative, what is the *least* number of factors in that product that *must be* negative? ## Footnote (a) none (b) 1 (c) 1,561 (d) all

Answer 69

(b) 1 ## Footnote Only an odd number of negative factors can produce a negative product. Therefore, *at least* one factor *must* be negative.

Question 70

The graph of *y* = 5 is perpendicular to the graph of? ## Footnote (a) *y* = 0 (b) *y* = *x* + 5 (c) *x* = 5

Answer 70

(c) x = 5

Question 71

If the coordinates of the vertices of a rectangle are A(2, 0), B(2, 4), C(5, 4), D(5, 0), what is the distance between the vertices that lay on its diagonal?

Answer 71

The distance between two points on a diagonal of the rectangle ABCD is 5. The diagonal vertices are A and C or B and D. Use **distance formula** to answer the question without graphing the rectangle. ![]() Working with points A and C, you get (4 - 0)² + (5 - 2)² = 25. The square root of 25 equals 5. *d* = 5.

Question 72

Find the midpoint between two points with the coordinates (5, 7) and (-3, -5).

Answer 72

The coordinates of the midpoint are (1,1). You don't have to memorize the formula as long as you know the concept that finding the midpoint is like finding the average of the *x* and the *y* coordinates of the two points in question. So, ^{5 + (-3)}*/*₂ = 1 ^{7 + (-5)}*/*₂ = 1

Question 73

II*x*I + *y*I = 6 What is the largest possible value of *y*?

Answer 73

The largest possible value of *y* is 6. Write two equations for the absolute value. I*x*I + *y* = 6 ⇒ I*x*I = 6 - *y* I*x*I + *y* = -6 ⇒ I*x*I = -6 - *y* The absolute value of *x* must be greater or equal to zero. In the 1st equation, *y* could be as large as 6. In the 2nd equation, *y* could be as large as -6.

Question 74

If *A*(4, 6) is the midpoint of *DC* with *D* = (1, 2) and *C* = (*x, y*), what are the values of *x* and *y*?

Answer 74

The value of *x* is 7, the value of *y* is 10. Since *A* is the midpoint of *DC*, ^{(1 + x)}*/*₂ = 4 ^{(2 + y)}*/*₂ = 6 Now solve for *x* and *y*.

Question 75

What is true about the slopes of two perpendicular lines? (excluding vertical and horizontal lines) ## Footnote (a) their quotient equals 1 (b) their product equals -1 (c) their absolute values are equal (d) none of the above (e) all of the above

Answer 75

(b) The product of the slopes of two perpendicular lines equals -1. ## Footnote The slopes of perpendicular lines are the opposite reciprocal of each other.

Question 76

What is true about the slopes of two parallel lines? (excluding vertical and horizontal lines) ## Footnote (a) their quotient equals 1 (b) their product equals -1 (c) their absolute values are equal (d) both (a) and (c) (e) all of the above

Answer 76

(d) both (a) and (c) ## Footnote The slopes of parallel lines are equal so their quotient equals 1. Naturally, the same numbers have the same absolute values.

Question 77

If given an equation of a line, how do you find its slope? *ax + ky = c*

Answer 77

To find the slope of a line from an equation, put the equation into the *slope-intercept form*. ***y = mx + b*** where *m* is the slope * ax + ky = c* * ky = c - ax* ***y = ^c/_k - ^a/_kx*** This is the slope-intercept form of the standard form of the equation. The slope equals - *^a/_k*.

Question 78

Which of the following lines is parallel to the line y = 7 - 5*x*? * I. y = 7 + ¹*/*₅*x* * II. y = -7 - 5*x* * III. y = 7 + 5*x* (a) I only (b) II only (c) III only (d) none of the above

Answer 78

(b) II only ## Footnote The slopes of parallel lines are the same. Only II has the same slope as the line given in the question stem.

Question 79

At what point does the line described by the equation below cross y-axis? 10*x -* 5*y =* 25 (a) (5, 0) (b) (0, 5) (c) (25, 5) (d) (0, -5) (e) (0, 10)

Answer 79

(d) (0, -5) ## Footnote You need to find the *y*-intercept of the line so put the equation in the slope-intercept form. 10*x* - 25 = 5y ⇒ 2*x* - 5 = *y* Now, it's clear that the *y*-intercept of this line has the coordinates (0, -5). \*\*\* The x-coordinate of y-intercept is always zero so eliminate (a) and (c) right away.

Question 80

What is (are) the value(s) of *x* if I2*x* - 5I = 11 (a) 8 or 3 (b) -3 (c) 8 or -3 (d) -8 or -3 (e) 8

Answer 80

(c) 8 or -3 ## Footnote By definition of the absolute value, since I2x - 5I = 11, then 2x - 5 can equal 11 or -11. Now, solve: 2x - 5 = 11 ⇒2x = 16 ⇒ x = 8 2x - 5 = -11 ⇒ 2x = -6 ⇒ x = -3

Question 81

What is(are) the value(s) of 3*x* if 6 + I2*x* - 5I = 11 and *x* is positive?

Answer 81

*x* = 15 ## Footnote First, isolate the absolute value in the equation by subtracting 6 from each side. I2x - 5I = 5 By definition of the absolute value, since I2x - 5I = 5, then 2x - 5 can equal 5 or -5. Now, solve: 2x - 5 = 5 ⇒2x = 10 ⇒ x = 5 2x - 5 = -5 ⇒ 2x = 0 ⇒ x = 0 Remember to pick only the positive value of *x*.

Question 82

What is one possible value of *x* if I*x* - 5I = I-11I? ## Footnote (a) -6 (b) -11 (c) -16 (d) -21

Answer 82

(a) *x* = -6 ## Footnote The absolute value of -11 is 11. By definition of the absolute value, since I*x* - 5I = 11, then *x* - 5 can equal 11 or -11. Now, solve: *x* - 5 = 11 ⇒*x* = 16 x - 5 = -11 ⇒ x = -6 16 is not among the answer choices, so choice (a) -6 is the correct answer.