1.1 Real Numbers: Algebra Essentials

Write each of the following as a rational number.

1. ⓐ7
2. ⓑ0
3. ⓒ–8

Write each of the following as a rational number.

1. ⓐ11
2. ⓑ3
3. ⓒ–4

Write each of the following rational numbers as either a terminating or repeating decimal.

1. ⓐ $- \frac{5}{7}$
2. ⓑ $\frac{15}{5}$
3. ⓒ $\frac{13}{25}$

Write each of the following rational numbers as either a terminating or repeating decimal.

1. ⓐ $\frac{68}{17}$
2. ⓑ $\frac{8}{13}$
3. ⓒ $- \frac{17}{20}$

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

1. ⓐ $\sqrt{25}$
2. ⓑ $\frac{33}{9}$
3. ⓒ $\sqrt{11}$
4. ⓓ $\frac{17}{34}$
5. ⓔ $0.3033033303333 \ldots$

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

1. ⓐ $\frac{7}{77}$
2. ⓑ $\sqrt{81}$
3. ⓒ $4.27027002700027 \ldots$
4. ⓓ $\frac{91}{13}$
5. ⓔ $\sqrt{39}$

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. ⓐ $- \frac{10}{3}$
2. ⓑ $\sqrt{5}$
3. ⓒ $- \sqrt{289}$
4. ⓓ $−6 \pi$
5. ⓔ $0.615384615384 \ldots$

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. ⓐ $\sqrt{73}$
2. ⓑ $−11.411411411 \ldots$
3. ⓒ $\frac{47}{19}$
4. ⓓ $- \frac{\sqrt{5}}{2}$
5. ⓔ $6.210735$

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

1. ⓐ $\sqrt{36}$
2. ⓑ $\frac{8}{3}$
3. ⓒ $\sqrt{73}$
4. ⓓ $−6$
5. ⓔ $3.2121121112 \ldots$

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

1. ⓐ $- \frac{35}{7}$
2. ⓑ $0$
3. ⓒ $\sqrt{169}$
4. ⓓ $\sqrt{24}$
5. ⓔ $4.763763763 \ldots$

Use the order of operations to evaluate each of the following expressions.

1. ⓐ $\left(3 \cdot 2\right)^{2} - 4 \left(6 + 2\right)$
2. ⓑ $\frac{5^{2} - 4}{7} - \sqrt{11 - 2}$
3. ⓒ $6 - \left|5 - 8\right| + 3 \left(4 - 1\right)$
4. ⓓ $\frac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}}$
5. ⓔ $7 \left(5 \cdot 3\right) - 2 \left[\left(6 - 3\right) - 4^{2}\right] + 1$

Use the order of operations to evaluate each of the following expressions.

1. ⓐ $\sqrt{5^{2} - 4^{2}} + 7 \left(5 - 4\right)^{2}$
2. ⓑ $1 + \frac{7 \cdot 5 - 8 \cdot 4}{9 - 6}$
3. ⓒ $\left|1.8 - 4.3\right| + 0.4 \sqrt{15 + 10}$
4. ⓓ $\frac{1}{2} \left[5 \cdot 3^{2} - 7^{2}\right] + \frac{1}{3} \cdot 9^{2}$
5. ⓔ $\left[\left(3 - 8\right)^{2} - 4\right] - \left(3 - 8\right)$

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

1. ⓐ $3 \cdot 6 + 3 \cdot 4$
2. ⓑ $\left(5 + 8\right) + \left(−8\right)$
3. ⓒ $6 - \left(15 + 9\right)$
4. ⓓ $\frac{4}{7} \cdot \left(\frac{2}{3} \cdot \frac{7}{4}\right)$
5. ⓔ $100 \cdot \left[0.75 + \left(−2.38\right)\right]$

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

1. ⓐ $\left(- \frac{23}{5}\right) \cdot \left[11 \cdot \left(- \frac{5}{23}\right)\right]$
2. ⓑ $5 \cdot \left(6.2 + 0.4\right)$
3. ⓒ $18 - \left(7 −15\right)$
4. ⓓ $\frac{17}{18} + \left[\frac{4}{9} + \left(- \frac{17}{18}\right)\right]$
5. ⓔ $6 \cdot \left(−3\right) + 6 \cdot 3$

List the constants and variables for each algebraic expression.

1. ⓐ x + 5
2. ⓑ $\frac{4}{3} \pi r^{3}$
3. ⓒ $\sqrt{2 m^{3} n^{2}}$

List the constants and variables for each algebraic expression.

1. ⓐ $2 \pi r \left(r + h\right)$
2. ⓑ2(L + W)
3. ⓒ $4 y^{3} + y$

Evaluate the expression $2 x - 7$ for each value for x.

1. ⓐ $x = 0$
2. ⓑ $x = 1$
3. ⓒ $x = \frac{1}{2}$
4. ⓓ $x = −4$

Evaluate the expression $11 - 3 y$ for each value for y.

1. ⓐ $y = 2$
2. ⓑ $y = 0$
3. ⓒ $y = \frac{2}{3}$
4. ⓓ $y = −5$

Evaluate each expression for the given values.

1. ⓐ $x + 5$ for $x = −5$
2. ⓑ $\frac{t}{2 t −1}$ for $t = 10$
3. ⓒ $\frac{4}{3} \pi r^{3}$ for $r = 5$
4. ⓓ $a + a b + b$ for $a = 11 , b = −8$
5. ⓔ $\sqrt{2 m^{3} n^{2}}$ for $m = 2 , n = 3$

Evaluate each expression for the given values.

1. ⓐ $\frac{y + 3}{y - 3}$ for $y = 5$
2. ⓑ $7 - 2 t$ for $t = −2$
3. ⓒ $\frac{1}{3} \pi r^{2}$ for $r = 11$
4. ⓓ $\left(p^{2} q\right)^{3}$ for $p = −2 , q = 3$
5. ⓔ $4 \left(m - n\right) - 5 \left(n - m\right)$ for $m = \frac{2}{3} , n = \frac{1}{3}$

Simplify each algebraic expression.

1. ⓐ $3 x - 2 y + x - 3 y - 7$
2. ⓑ $2 r - 5 \left(3 - r\right) + 4$
3. ⓒ $\left(4 t - \frac{5}{4} s\right) - \left(\frac{2}{3} t + 2 s\right)$
4. ⓓ $2 m n - 5 m + 3 m n + n$

Simplify each algebraic expression.

1. ⓐ $\frac{2}{3} y - 2 \left(\frac{4}{3} y + z\right)$
2. ⓑ $\frac{5}{t} - 2 - \frac{3}{t} + 1$
3. ⓒ $4 p \left(q - 1\right) + q \left(1 - p\right)$
4. ⓓ $9 r - \left(s + 2 r\right) + \left(6 - s\right)$

A rectangle with length $L$ and width $W$ has a perimeter $P$ given by $P = L + W + L + W .$ Simplify this expression.

If the amount $P$ is deposited into an account paying simple interest $r$ for time $t ,$ the total value of the deposit $A$ is given by $A = P + P r t .$ Simplify the expression. (This formula will be explored in more detail later in the course.)

Is $\sqrt{2}$ an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

Simplify the given expression.

$6 \div 2 - \left(81 \div 3^{2}\right)$

Simplify the given expression.

$−2 \times \left[16 \div \left(8 - 4\right)^{2}\right]^{2}$

Simplify the given expression.

$3 \left(5 - 8\right)$

Simplify the given expression.

$12 \div \left(36 \div 9\right) + 6$

Simplify the given expression.

$3 - 12 \times 2 + 19$

Simplify the given expression.

$5 + \left(6 + 4\right) - 11$

Simplify the given expression.

$14 \times 3 \div 7 - 6$

Simplify the given expression.

$6 + 2 \times 2 - 1$

Simplify the given expression.

$9 + 4 \left(2^{2}\right)$

Simplify the given expression.

$25 \div 5^{2} - 7$

Simplify the given expression.

$2 \times 4 - 9 \left(−1\right)$

Simplify the given expression.

$12 \left(3 - 1\right) \div 6$

Evaluate the expression using the given value of the variable.

$4 y + 8 – 2 y$ for $y = 3$

Evaluate the expression using the given value of the variable.

$4 z - 2 z \left(1 + 4\right) – 36$ for $z = 5$

Evaluate the expression using the given value of the variable.

$- \left(2 x\right)^{2} + 1 + 3$ for $x = 2$

Evaluate the expression using the given value of the variable.

$2 \left(11 c - 4\right) – 36$ for $c = 0$

Evaluate the expression using the given value of the variable.

$\frac{1}{4} \left(8 w - 4^{2}\right)$ for $w = 1$

Simplify the expression.

$2 y - \left(4\right)^{2} y - 11$

Simplify the expression.

$8 b - 4 b \left(3\right) + 1$

Simplify the expression.

$7 z - 3 + z \times 6^{2}$

Simplify the expression.

$9 \left(y + 8\right) - 27$

Simplify the expression.

$6 + 12 b - 3 \times 6 b$

Simplify the expression.

$\left(\frac{4}{9}\right)^{2} \times 27 x$

Simplify the expression.

$9 x + 4 x \left(2 + 3\right) - 4 \left(2 x + 3 x\right)$

Consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

Solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by $\pi .$ Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Consider this scenario: There is a mound of $g$ pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budg…

Use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

$\left(0.25 - 0.75\right)^{2} x - 7.2 = 9.9$

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the simplified expression is rational or irrational: $\sqrt{−18 - 4 \left(5\right) \left(−1\right)} .$

The division of two natural numbers will always result in what type of number?