Question

1. do you think prokaryotic or eukaryotic organisms were the first organisms to inhabit earth? justify your answer.

name: jason spence
date: \tbell: \tunit 2: equations & inequalities homework 5: absolute value equations
this is a 2 - page document!
directions: solve each equation. show all steps!

1. |x| = 3
2. |m|=-10
3. |-6k| = 30
4. |c - 10| = 3
5. |2k + 3| = 7
6. |-4g - 2| = 10
7. |\frac{1}{2}x + 5| = 13
8. |26 - 4c|=-2

Explanation:

Step1: Recall absolute - value property

The absolute - value of a number \(|a|\) is defined as \(|a|=

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\). For the equation \(|x| = k\), if \(k>0\), then \(x = k\) or \(x=-k\); if \(k = 0\), then \(x = 0\); if \(k<0\), the equation has no solution since the absolute - value of a real number is non - negative.

Step2: Solve \(|x| = 3\)

Since \(3>0\), by the absolute - value property, \(x = 3\) or \(x=-3\).

Step3: Solve \(|m|=-10\)

Since \(-10<0\), the equation \(|m|=-10\) has no solution, denoted as \(\varnothing\).

Step4: Solve \(|-6k| = 30\)

First, note that \(|-6k|=|6k|\). Since \(30>0\), we have \(6k = 30\) or \(6k=-30\). Dividing both sides of \(6k = 30\) by 6 gives \(k = 5\), and dividing both sides of \(6k=-30\) by 6 gives \(k=-5\).

Step5: Solve \(|c - 10| = 3\)

If \(|c - 10| = 3\), then \(c-10 = 3\) or \(c - 10=-3\). Solving \(c-10 = 3\) by adding 10 to both sides gives \(c = 13\), and solving \(c - 10=-3\) by adding 10 to both sides gives \(c = 7\).

Step6: Solve \(|2k + 3| = 7\)

We have \(2k+3 = 7\) or \(2k + 3=-7\). For \(2k+3 = 7\), subtracting 3 from both sides gives \(2k = 4\), then dividing by 2 gives \(k = 2\). For \(2k+3=-7\), subtracting 3 from both sides gives \(2k=-10\), then dividing by 2 gives \(k=-5\).

Step7: Solve \(|-4g - 2| = 10\)

We have \(-4g-2 = 10\) or \(-4g - 2=-10\). For \(-4g-2 = 10\), adding 2 to both sides gives \(-4g = 12\), then dividing by \(-4\) gives \(g=-3\). For \(-4g - 2=-10\), adding 2 to both sides gives \(-4g=-8\), then dividing by \(-4\) gives \(g = 2\).

Step8: Solve \(|\frac{1}{2}x+5| = 13\)

We have \(\frac{1}{2}x+5 = 13\) or \(\frac{1}{2}x+5=-13\). For \(\frac{1}{2}x+5 = 13\), subtracting 5 from both sides gives \(\frac{1}{2}x = 8\), then multiplying by 2 gives \(x = 16\). For \(\frac{1}{2}x+5=-13\), subtracting 5 from both sides gives \(\frac{1}{2}x=-18\), then multiplying by 2 gives \(x=-36\).

Step9: Solve \(|26 - 4c|=-2\)

Since \(-2<0\), the equation \(|26 - 4c|=-2\) has no solution, denoted as \(\varnothing\).

Prokaryotic organisms are thought to be the first. The Earth's early atmosphere was anaerobic. Prokaryotes, such as archaea and bacteria, are simpler in structure without a nucleus and can survive in harsh anaerobic conditions. Eukaryotic cells are more complex, likely evolving from prokaryotes through endosymbiosis. Fossil evidence also supports that prokaryotic - like fossils are older than eukaryotic ones.

Answer:

1. \(x = 3,x=-3\)
2. \(\varnothing\)
3. \(k = 5,k=-5\)
4. \(c = 13,c = 7\)
5. \(k = 2,k=-5\)
6. \(g=-3,g = 2\)
7. \(x = 16,x=-36\)
8. \(\varnothing\)

For the biological question "Do you think prokaryotic or eukaryotic organisms were the first organisms to inhabit Earth? Justify your answer.":