I need number 8 I didn’t get it. Please use an explanation.

The line segment FG represent the volume of water decreases in Katherine's water bottle.

From the given graph, x-axis represents the distance from home (km) and the y-axis represents the volume (L).

The line segment FG represent the volume of water decreases in Katherine's water bottle, the line segment HI represent the volume of water increases in Katherine's water bottle and the line segment KL represents there is no change in water level.

Therefore, the line segment FG represent the volume of water decreases in Katherine's water bottle.

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The incorrect statement is:

B. The three-part inequality - 5x < 15x^2 < 3 is equivalent to - 6 ≤ 5 - x < 6.

In the given statement, there is an error in the inequality. The correct statement should be:

The three-part inequality - 5x < 15x^2 < 3 is equivalent to - 6 ≤ 5 - x and 5 - x < 6.

When solving the three-part inequality - 5x < 15x^2 < 3, we need to split it into two separate inequalities. The correct splitting should be:

- 5x < 15x^2 and 15x^2 < 3

Simplifying the first inequality:

- 5x < 15x^2

Dividing by x (assuming x ≠ 0), we need to reverse the inequality sign:

- 5 < 15x

Simplifying the second inequality:

15x^2 < 3

Dividing by 15, we get:

x^2 < 1/5

Taking the square root (assuming x ≥ 0), we have two cases:

x < 1/√5 and -x < 1/√5

Combining these inequalities, we get:

- 5 < 15x and x < 1/√5 and -x < 1/√5

Therefore, the correct statement is that the three-part inequality - 5x < 15x^2 < 3 is equivalent to - 6 ≤ 5 - x and 5 - x < 6, not - 6 ≤ 5 - x < 6 as stated in option B.

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Two roots of the equation are,

⇒x = 4.55, - 0.55

We have to given that,

A curve has the equation,

⇒ y = 2x² - 8x - 5

Now, We know that,

If the curve intercept at x - axis, then y value is zero.

Hence, We get;

⇒ y = 2x² - 8x - 5

⇒ 0 = 2x² - 8x - 5

⇒ 2x² - 8x - 5 = 0

By quadratic formula, we get;

⇒ x = (- (- 8)) ± √(- 8)² - 4×2×- 5) / 2×2

⇒ x = (8 ± √64 + 40) / 4

⇒ x = (8 ± 10.2) / 4

Take positive sign,

⇒x = (8 + 10.2) / 4

⇒ x = 18.2/4

⇒ x = 4.55

Take negative sign,

⇒ x = (8 - 10.2) / 4

⇒ x = - 2.2/4

⇒ x = - 0.55

Hence, Two roots of the equation are,

⇒x = 4.55, - 0.55

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y = 2sin(2πx) + 1 is an amplitude-scaled, period-halved, vertically shifted version of y = sin(x). It is important to note that y = cos(x) is not directly related to y = 2sin(2πx) + 1 as it represents a different trigonometric function.

The equation y = 2sin(2πx) + 1 is related to y = sin(x) and y = cos(x) through trigonometric transformations. Let's break down the relationship:

Amplitude: The coefficient 2 in front of sin(2πx) in y = 2sin(2πx) + 1 indicates that the amplitude of the sine function has been doubled compared to y = sin(x). The amplitude determines the maximum and minimum values of the function.

Period: In y = 2sin(2πx) + 1, the argument of sin(2πx) is 2πx, resulting in a period that is halved compared to y = sin(x). The period is the length of one complete cycle of the function.

Phase shift: There is no phase shift in y = 2sin(2πx) + 1. The function y = sin(x) has a phase shift of 0.

Vertical shift: The constant term +1 in y = 2sin(2πx) + 1 represents a vertical shift upward by 1 unit compared to y = sin(x). It shifts the entire graph vertically.

Overall, y = 2sin(2πx) + 1 is an amplitude-scaled, period-halved, vertically shifted version of y = sin(x). It is important to note that y = cos(x) is not directly related to y = 2sin(2πx) + 1 as it represents a different trigonometric function.

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