Absolute Value worksheet for practicing finding absolute values, comparing numbers, and solving absolute value equations.

Problem: Solving the Absolute Value Worksheet

#### Part 1: Find the absolute values.

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of any number is always non-negative.

- a. \( |-4| \)
The absolute value of \(-4\) is \(4\).
Answer: \(4\)

- b. \( |26| \)
The absolute value of \(26\) is \(26\).
Answer: \(26\)

- c. \( |-18| \)
The absolute value of \(-18\) is \(18\).
Answer: \(18\)

- d. \( |-3| \)
The absolute value of \(-3\) is \(3\).
Answer: \(3\)

- e. \( |-44| \)
The absolute value of \(-44\) is \(44\).
Answer: \(44\)

- f. \( |65| \)
The absolute value of \(65\) is \(65\).
Answer: \(65\)

- g. \( |-99| \)
The absolute value of \(-99\) is \(99\).
Answer: \(99\)

- h. \( |-6| \)
The absolute value of \(-6\) is \(6\).
Answer: \(6\)

- i. \( |-128| \)
The absolute value of \(-128\) is \(128\).
Answer: \(128\)

- j. \( |28| \)
The absolute value of \(28\) is \(28\).
Answer: \(28\)

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#### Part 2: Compare using \(<\), \(>\), or \(=\).

We compare the absolute values or the given numbers as required.

- k. \( |-4| \) _______ \( |-5| \)
\( |-4| = 4 \) and \( |-5| = 5 \).
Since \(4 < 5\), we have \( |-4| < |-5| \).
Answer: \(<\)

- l. \( 17 \) _______ \( |-17| \)
\( |-17| = 17 \).
Since \(17 = 17\), we have \( 17 = |-17| \).
Answer: \(=\)

- m. \( |29| \) _______ \( |-29| \)
\( |29| = 29 \) and \( |-29| = 29 \).
Since \(29 = 29\), we have \( |29| = |-29| \).
Answer: \(=\)

- n. \( 58 \) _______ \( |-59| \)
\( |-59| = 59 \).
Since \(58 < 59\), we have \( 58 < |-59| \).
Answer: \(<\)

- o. \( 30 \) _______ \( |-28| \)
\( |-28| = 28 \).
Since \(30 > 28\), we have \( 30 > |-28| \).
Answer: \(>\)

- p. \( |-7| \) _______ \( 0 \)
\( |-7| = 7 \).
Since \(7 > 0\), we have \( |-7| > 0 \).
Answer: \(>\)

- q. \( |86| \) _______ \( |-68| \)
\( |86| = 86 \) and \( |-68| = 68 \).
Since \(86 > 68\), we have \( |86| > |-68| \).
Answer: \(>\)

- r. \( |14| \) _______ \( -14 \)
\( |14| = 14 \).
Since \(14 > -14\), we have \( |14| > -14 \).
Answer: \(>\)

- s. \( |-156| \) _______ \( |-165| \)
\( |-156| = 156 \) and \( |-165| = 165 \).
Since \(156 < 165\), we have \( |-156| < |-165| \).
Answer: \(<\)

- t. \( |3| \) _______ \( |-3| \)
\( |3| = 3 \) and \( |-3| = 3 \).
Since \(3 = 3\), we have \( |3| = |-3| \).
Answer: \(=\)

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#### Part 3: Write two solutions for each variable in the equations below.

The equation \( |x| = a \) has two solutions: \( x = a \) and \( x = -a \).

- u. \( |\alpha| = 7 \)
The solutions are \( \alpha = 7 \) and \( \alpha = -7 \).
Answer: \( \alpha = 7 \) and \( \alpha = -7 \)

- v. \( |\rho| = 41 \)
The solutions are \( \rho = 41 \) and \( \rho = -41 \).
Answer: \( \rho = 41 \) and \( \rho = -41 \)

- w. \( |y| = 256 \)
The solutions are \( y = 256 \) and \( y = -256 \).
Answer: \( y = 256 \) and \( y = -256 \)

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Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{a. } 4 & \text{b. } 26 \\
\text{c. } 18 & \text{d. } 3 \\
\text{e. } 44 & \text{f. } 65 \\
\text{g. } 99 & \text{h. } 6 \\
\text{i. } 128 & \text{j. } 28 \\
\text{k. } < & \text{l. } = \\
\text{m. } = & \text{n. } < \\
\text{o. } > & \text{p. } > \\
\text{q. } > & \text{r. } > \\
\text{s. } < & \text{t. } = \\
\text{u. } \alpha = 7 \text{ and } \alpha = -7 & \\
\text{v. } \rho = 41 \text{ and } \rho = -41 & \\
\text{w. } y = 256 \text{ and } y = -256 &
\end{array}
}
\]