# 1.1 Real numbers: algebra essentials  (Page 10/35)

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$4y+8=2y$

$-4$

$\left(11a+3\right)-18a=-4$

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

$-6$

$4y{\left(7-2\right)}^{2}=-200$

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

$±1$

$8\left(2+4\right)-15b=b$

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

2

$4\left(3-1\right)x=4$

$\frac{1}{4}\left(8w-{4}^{2}\right)=0$

2

For the following exercises, simplify the expression.

$4x+x\left(13-7\right)$

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

$-14y-11$

$\frac{a}{{2}^{3}}\left(64\right)-12a÷6$

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

$-4b+1$

$5l÷3l\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\left(9-6\right)$

$7z-3+z\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{6}^{2}$

$43z-3$

$4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3+18x÷9-12$

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

$9y+45$

$\left(\frac{9}{6}t-4\right)2$

$6+12b-3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}6b$

$-6b+6$

$18y-2\left(1+7y\right)$

${\left(\frac{4}{9}\right)}^{2}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}27x$

$\frac{16x}{3}$

$8\left(3-m\right)+1\left(-8\right)$

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

$9x$

${5}^{2}-4\left(3x\right)$

## Real-world applications

For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends$10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighbor’s car. Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations. $\frac{1}{2}\left(40-10\right)+5$ How much money does Fred keep? For the following exercises, 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 $\text{\hspace{0.17em}}\pi .\text{\hspace{0.17em}}$ Is the circumference of a quarter a whole number, a rational number, or an irrational number? irrational number Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact? For the following exercises, consider this scenario: There is a mound of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ 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. $g+400-2\left(600\right)=1200$ Solve for g . For the following exercise, solve the given problem. Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got$2.5 million for the annual marketing budget. He must spend the budget such that $\text{\hspace{0.17em}}2,500,000-x=0.\text{\hspace{0.17em}}$ What property of addition tells us what the value of x must be?

## Technology

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

$0.5{\left(12.3\right)}^{2}-48x=\frac{3}{5}$

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

68.4

## Extensions

If a whole number is not a natural number, what must the number be?

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

true

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: $\text{\hspace{0.17em}}\sqrt{-18-4\left(5\right)\left(-1\right)}.$

irrational

Determine whether the simplified expression is rational or irrational: $\text{\hspace{0.17em}}\sqrt{-16+4\left(5\right)+5}.$

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

rational

What property of real numbers would simplify the following expression: $\text{\hspace{0.17em}}4+7\left(x-1\right)?$

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