Equations, Expressions, and Inequalities Q&A: Get Expert Help and Learn from Real-World Examples

The model for radioactive decay is ${\displaystyle y=y0e^{-kt}}$. A radioactive substance has a half-life of 1150 years. If 15 grams are present today, in how many years will 10 grams be present? (Round the value of k to 7 decimal places.) Round your answer to two decimal places, if necessary.

To choose: The correct option that describes a from the given choices, if a is a nonnegative real number.
Given:
a. ${\displaystyle a<0}$
b. ${\displaystyle a>0}$
c. ${\displaystyle a\le 0}$
d. ${\displaystyle a\ge 0}$

Write a, b, c, and d in least to greatest.
Given the inequality equation:
${\displaystyle b>d+\frac{1}{3};c+1<a-4;d+\frac{5}{8}>a+2}$

Min ${\displaystyle P=4x+10y}$
Subject to:
${\displaystyle 2x+6y\ge 24}$
${\displaystyle 4x+2y\ge 18}$

To calculate: The solution of absolute value inequality ${\displaystyle -9\le 4-|2k-1|}$ and write solution set in interval notation.

Solve and check the following equation.
${\displaystyle \frac{x-7}{5}+\frac{1}{5}=-\frac{x}{10}}$
Select the correct choice below and fill in any answer boxes in your choice.
A. The solution set is (_).
B. The solution is the empty set.

Simplify the following expressions using the imaginary number i:
1. ${\displaystyle 5\sqrt{-12}}$
2. ${\displaystyle \sqrt{-8}}$
3. ${\displaystyle 3\sqrt{-7}}$
4. ${\displaystyle -3\sqrt{-200}}$
Write the following numbers using the imaginary number i, and then perform the operations necessary and simplify your answer.
1. ${\displaystyle 3\sqrt{-9}\cdot 2\sqrt{-49}}$
2. ${\displaystyle \sqrt{-12}\cdot \sqrt{-36}}$
3. ${\displaystyle i\sqrt{-36}-i2}$
Simplify the following expressions using the imaginary number i when needed:
8. i8
9. i75
10. (i2)(i6)
Simplify and rationalize the denominators.
11. ${\displaystyle 4\sqrt{-\frac{3}{2}}}$
12. ${\displaystyle -3\sqrt{-\frac{2}{27}}}$
13. ${\displaystyle \sqrt{-\frac{4}{27}}}$
Perform the operation indicated in each problem below:
14. ${\displaystyle (8+6i)+(2+i)}$
15. ${\displaystyle -9+(4-2i)}$
16. ${\displaystyle (1+i)-(1-i)}$
17. ${\displaystyle 3i-(4+15i)}$
Perform the operation indicated in each problem below:
18. ${\displaystyle (\sqrt{2}+3i)(\sqrt{2}-3i)}$
19. ${\displaystyle (\sqrt{27}-3i)(\sqrt{3}+i)}$
20. ${\displaystyle (\sqrt{2}-i\sqrt{3})(\sqrt{2}+i\sqrt{3})}$
21. ${\displaystyle 2i(3i-2)}$
22. ${\displaystyle (3-6i)2}$
Determine the sum or difference in geometric terms, and then check it algebraically. Remember that ${\displaystyle a-b=a+\left(b\right)}$.
23. ${\displaystyle (8-5i)-(2+3i)}$
24. ${\displaystyle (-6+i)-(-2-3i)}$
25. ${\displaystyle (3+i)+(-2+2i)+(8-2i)}$
Solve the following equations using the quadratic formula.
26. ${\displaystyle X2-4X+13=0}$
27. ${\displaystyle X2+49=0}$
28. ${\displaystyle X2+5X+8=0}$
29. ${\displaystyle X2-3X+4=0}$

Explain in your own words what it means for an equation to model a real-world situation, and give an example.

What is wrong with the following equation: ${\displaystyle 4+\left(\frac{10}{5}\right)=6-2}$
A) Equations should not involve expressions on both sides.
B)The equation mixes different operations like +, / and -.
C) The left sides

Consider the following statement: Mary wants to (spend, at least spend, at most spend) 20 minutes a day exercising. Depending on which word you chose in parenthesis you end up with a different expression.
Write the expression you would get for each of the word choices and label them.

Statement showing a relationship between numbers that are not necessarily equal using the symbols ${\displaystyle >,<,\le ,\ge ,\text{or}\ne q}$ are called _.
1) inferences
2) inconsistencies
3) unequals
4) inequalities

Find the domain of the function.
${\displaystyle f\left(x\right)=\sqrt{x+4}-\frac{\sqrt{1-x}}{x}}$

A. Characterized the x-intercept and the y-intercept of the given linear equations below. Then use them to draw and characterized the graph of the linear equation.
Note: characterized means to tell whether the slope of the line is positive or negative and to tell whether the line is pointing upward to the right or downward to the right.
1. ${\displaystyle -6x+3y=-18}$
2. ${\displaystyle 3x+2y=9x+12}$

The solution set of the equation ${\displaystyle {(2y-3)}^{\frac{1}{3}}-{(4y+5)}^{\frac{1}{3}}=0}$

The function ${\displaystyle 2(4x+1)}$ and ${\displaystyle 8x+2}$ are either equivalent inequalities, equivalent equation, equivalent expression, or not equivalent.

If ${\displaystyle P\left(0<z<k\right)=.3212,}$ find and solve k.

Solve this equation ${\displaystyle \frac{2x}{x+2}=x–1}$

Identify, whether ${\displaystyle -\frac{1}{2}x<7,x>14}$ are equivalent inequalities or not equivalent ones?

Table shows the number of cellular phone subscribers in the United States and their average monthly bill in the years from 2000 to 2010.
$\begin{array}{ccc}\text{Year}& \text{Subscribers}& \text{Average Local}\\ & \text{(millions)}& \text{Monthly Bill ()}\\ \text{2000}& \text{109.5}& \text{45.27}\\ \text{2001}& \text{128.4}& \text{47.37}\\ \text{2002}& \text{140.8}& \text{48.40}\\ \text{2003}& \text{158.7}& \text{49.91}\\ \text{2004}& \text{182.1}& \text{50.64}\\ \text{2005}& \text{207.9}& \text{49.98}\\ \text{2006}& \text{233.0}& \text{50.56}\\ \text{2007}& \text{255.4}& \text{49.79}\\ \text{2008}& \text{262.7}& \text{50.07}\\ \text{2009}& \text{276.6}& \text{48.16}\\ \text{2010}& \text{300.5}& \text{47.21}\end{array}$
In 1995 There were 33.8 million subscribers with an average local monthly bill of $51.00. Add these points to the scatter plots. Do the 1995 points match well with the trends in the rest of the data?

Solve the equation ${\displaystyle x^3–5x^2+2x+8=0}$ given that -1 is a zero of ${\displaystyle f\left(x\right)=x^3–5x^2+2x+8}$