# 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?

inverse property of addition

## 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: $\text{\hspace{0.17em}}4+7\left(x-1\right)?$

#### Questions & Answers

How look for the general solution of a trig function
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why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
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what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as