Savanah solved the equation 3 + 4|3 plus 4 StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 11. + 3| = 11 for one solution. Her work is shown below:

Question

4|5 lines of math. Line 1 is, 4 StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 8. Line 2 is, StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 2. LIne 3 is, StartFraction x Over 2 EndFraction plus 3 equals 2. Line 4 is, StartFraction x Over 2 EndFraction equals negative 1. Line 5 is x equals negative 2. + 3| = 8
|5 lines of math. Line 1 is, 4 StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 8. Line 2 is, StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 2. LIne 3 is, StartFraction x Over 2 EndFraction plus 3 equals 2. Line 4 is, StartFraction x Over 2 EndFraction equals negative 1. Line 5 is x equals negative 2. + 3| = 2
5 lines of math. Line 1 is, 4 StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 8. Line 2 is, StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 2. LIne 3 is, StartFraction x Over 2 EndFraction plus 3 equals 2. Line 4 is, StartFraction x Over 2 EndFraction equals negative 1. Line 5 is x equals negative 2. + 3 = 2
5 lines of math. Line 1 is, 4 StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 8. Line 2 is, StartAbsoluteValue StartFraction x Over 2 EndFraction plus 3 EndAbsoluteValue equals 2. LIne 3 is, StartFraction x Over 2 EndFraction plus 3 equals 2. Line 4 is, StartFraction x Over 2 EndFraction equals negative 1. Line 5 is x equals negative 2. = –1
x = –2
What is the other solution to the given absolute value equation?

x =

Answer 1

Let's solve the given equation step by step starting from the beginning:

1. Original Equation:
   \[ 3 + 4 \left| \frac{x}{2} \right| + 3 = 11 \]

2. Simplify:
   Combine like terms: \[ 4 \left| \frac{x}{2} \right| + 6 = 11 \]

3. Isolate the Absolute Value:
   Subtract 6 from both sides: \[ 4 \left| \frac{x}{2} \right| = 5 \]

4. Divide by 4:
   \[ \left| \frac{x}{2} \right| = \frac{5}{4} \]

5. Remove the Absolute Value:
   This gives us two equations: \[ \frac{x}{2} = \frac{5}{4} \quad \text{and} \quad \frac{x}{2} = -\frac{5}{4} \]

6. Solve for x in Each Case:

   • First Case: \[ \frac{x}{2} = \frac{5}{4} \implies x = \frac{5}{4} \times 2 = \frac{10}{4} = \frac{5}{2} \]
   • Second Case: \[ \frac{x}{2} = -\frac{5}{4} \implies x = -\frac{5}{4} \times 2 = -\frac{10}{4} = -\frac{5}{2} \]

Thus, the solutions to the given absolute value equation are:

\[ x = \frac{5}{2} \] and \[ x = -\frac{5}{2} \]

Therefore, the other solution to the given absolute value equation is:

\[ \boxed{-\frac{5}{2}} \]

Answer 2

its -10

Which statements are true for solving the equation 0.5 – |x – 12| = –0.25? Check all that apply.

The equation will have no solutions.
A good first step for solving the equation is to subtract 0.5 from both sides of the equation.
A good first step for solving the equation is to split it into a positive case and a negative case.
The positive case of this equation is 0.5 – |x – 12| = 0.25.
The negative case of this equation is x – 12 = –0.75.
The equation will have only 1 solution

Answer 3

Let's analyze the equation \(0.5 - |x - 12| = -0.25\) step by step and assess the given statements.

1. Start with the original equation: \[ 0.5 - |x - 12| = -0.25 \]

2. Subtract \(0.5\) from both sides: \[ -|x - 12| = -0.25 - 0.5 \] \[ -|x - 12| = -0.75 \] \[ |x - 12| = 0.75 \] This tells us that we need to find \(x\) such that the absolute value of \(x - 12\) equals \(0.75\).

3. Set up the cases: From \(|x - 12| = 0.75\), we derive two equations: \[ x - 12 = 0.75 \quad \text{(positive case)} \] \[ x - 12 = -0.75 \quad \text{(negative case)} \]

4. Solve each case:

   • Positive Case: \[ x - 12 = 0.75 \implies x = 12 + 0.75 = 12.75 \]
   • Negative Case: \[ x - 12 = -0.75 \implies x = 12 - 0.75 = 11.25 \]

So the solutions are \(x = 12.75\) and \(x = 11.25\). Now, let's evaluate the statements:

1. The equation will have no solutions.
   False: The equation does have solutions (where \(x = 12.75\) and \(x = 11.25\)).

2. A good first step for solving the equation is to subtract 0.5 from both sides of the equation.
   True: This is an appropriate first step, which we performed above.

3. A good first step for solving the equation is to split it into a positive case and a negative case.
   False: Although we eventually do split into cases, we first need to simplify the equation.

4. The positive case of this equation is \(0.5 – |x – 12| = 0.25\).
   False: The correct positive case comes from solving \(|x - 12| = 0.75\), not modifying the original equation directly.

5. The negative case of this equation is \(x - 12 = -0.75\).
   True: This is one of the correct cases derived from the absolute value equation.

6. The equation will have only 1 solution.
   False: The equation has 2 solutions (\(x = 12.75\) and \(x = 11.25\)).

Summary of True Statements:

• A good first step for solving the equation is to subtract 0.5 from both sides of the equation.
• The negative case of this equation is \(x - 12 = -0.75\).

So the true statements are:

• A good first step for solving the equation is to subtract 0.5 from both sides of the equation.
• The negative case of this equation is \(x - 12 = -0.75\).

Answer 4

Josefina started to solve for x in the equation

3.5 = 1.9 – 0.8|2x – 0.6| using the steps below:

3.5 = 1.9 – 0.8|2x – 0.6|
1.6 = –0.8|2x – 0.6|
–2 = |2x – 0.6|
Why did Josefina stop at Step 3?

Answer 5

Josefina stopped at Step 3 because the equation she arrived at, \(-2 = |2x - 0.6|\), is not valid for any real number values of \(x\).

The absolute value \( |2x - 0.6| \) represents the distance of \(2x - 0.6\) from zero on the number line, which is always a non-negative value (greater than or equal to zero). However, she found that the absolute value equals \(-2\), a negative number.

Since an absolute value cannot equal a negative number, this indicates that there are no solutions to the original equation, and thus, Josefina correctly recognizes that she cannot proceed further.

In summary, Josefina stopped because she realized that the expression for the absolute value can never be negative, indicating there are no valid solutions to the equation.

Answer 6

For nitrogen to be a liquid, its temperature must be within 12.78 °F of –333.22 °F. Which equation can be used to find the maximum and minimum temperatures at which nitrogen is a liquid, x?



The equation that can be used to find the maximum and minimum temperatures is

|x +
| =
.

Answer 7

To set up the equation that describes the temperature range for nitrogen to be a liquid, we can use the information you provided.

The critical temperature for nitrogen is \(-333.22 , \text{°F}\), and it must be within \(12.78 , \text{°F}\) of this temperature. This can be expressed mathematically using the absolute value.

The condition that \(x\) (the temperature we are trying to find) is within \(12.78\) °F of \(-333.22\) °F can be written as:

\[ |x + 333.22| = 12.78 \]

This equation means that the temperature \(x\) can be \(12.78\) °F above or below \(-333.22\) °F.

Now, if we rewrite this expression in terms of a single variable \(x\):

\[ |x - (-333.22)| = 12.78 \implies |x + 333.22| = 12.78 \]

Summary

Thus, the final equation that can be used to determine the maximum and minimum temperatures at which nitrogen is a liquid is:

\[ |x + 333.22| = 12.78 \]

You can now find the maximum and minimum temperatures by solving this absolute value equation, which will break down into two cases:

1. \(x + 333.22 = 12.78\)
2. \(x + 333.22 = -12.78\)

After calculating these, you will obtain the maximum and minimum temperatures for liquid nitrogen.

Answer 8

x+333.22=12.78

For nitrogen to be a liquid, its temperature must be within 12.78 °F of –333.22 °F. The equation |x + 333.22 | = 12.78 can be used to find x, which represents the maximum and minimum temperatures at which nitrogen is a liquid.



is the maximum temperature at which nitrogen is a liquid.

is the minimum temperature at which nitrogen is a liquid.

Answer 9

To find the maximum and minimum temperatures at which nitrogen is a liquid from the equation

\[ |x + 333.22| = 12.78, \]

we need to solve the absolute value equation, which leads to two separate cases.

Step 1: Set up the equations from the absolute value

1. Positive Case: \[ x + 333.22 = 12.78 \]

2. Negative Case: \[ x + 333.22 = -12.78 \]

Step 2: Solve each case

1. Positive Case: \[ x + 333.22 = 12.78 \] \[ x = 12.78 - 333.22 \] \[ x = -320.44 , \text{°F} \]

2. Negative Case: \[ x + 333.22 = -12.78 \] \[ x = -12.78 - 333.22 \] \[ x = -345.44 , \text{°F} \]

Summary of Results

• The maximum temperature at which nitrogen is a liquid is: \(-320.44 , \text{°F}\)

• The minimum temperature at which nitrogen is a liquid is: \(-345.44 , \text{°F}\)

In conclusion:

• Maximum temperature: \( -320.44 , \text{°F} \)
• Minimum temperature: \( -345.44 , \text{°F} \)

Answer 10

the first is -320.44 and 2nd is -346

Max rides his scooter toward Kim and then passes her at a constant speed. His distance in feet, d, from Kim t seconds after he started riding his scooter is given by d = |150 – 9t|.

What does the 150 in the equation represent?
1. maxs starting distance from kim, maxs constant speed, the time it takes max to reach kim

What does the 9 in the equation represent?
2. Maxs starting distance from kim, The time it takes max to reach kim, maxs constant speed

Answer 11

To understand the components of the equation \(d = |150 - 9t|\), let's break it down:

1. The 150 in the equation:

   • The value of 150 represents Max's starting distance from Kim. This is the initial distance (in feet) that Max is from Kim when he begins riding his scooter. When \(t = 0\), the equation evaluates to: \[ d = |150 - 9(0)| = |150| = 150 \] Therefore, the correct answer is:
   • Max's starting distance from Kim.

2. The 9 in the equation:

   • The value of 9 represents Max's constant speed in feet per second. It is the rate at which Max's distance from Kim changes over time. The term \(-9t\) means that for every second that passes, Max gets 9 feet closer to Kim (which is why the distance \(d\) decreases). Therefore, the correct answer is:
   • Max's constant speed.

Summary:

1. The 150 represents Max's starting distance from Kim.
2. The 9 represents Max's constant speed.