The fact that $2(x+3)=2 x+6$ is attributable to the ___________ Property.

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The fact that $3 x=0$ implies that $x=0$ is a result of the _____________ Property.

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The domain of the variable in the expression $\frac{x}{x-4}$ is __________.

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True or False

Multiplying both sides of an equation by any number results in an equivalent equation.

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An equation that is satisfied for every value of the variable for which both sides are defined is called a(n) ________.

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An equation of the form $a x+b=0$ is called a(n) _______ equation or a(n) ___________ equation

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Multiple Choice

An admissible value for the variable that makes the equation a true statement is called a(n) ______ of the equation.

(a) identity

(b) solution

(c) degree

(d) model

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Multiple Choice

A chemist mixes 10 liters of a $20 \%$ solution with $x$ liters of a $35 \%$ solution. Which of the following expressions represents the total number of liters in the mixture?

(a) $x$

(b) $20-x$

(c) $\frac{35}{x}$

(d) $10+x$

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Solve each equation, if possible.

$$\frac{3}{2} x+2=\frac{1}{2}-\frac{1}{2} x$$

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Solve each equation, if possible.

$$\frac{2}{3} p=\frac{1}{2} p+\frac{1}{3}$$

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Solve each equation, if possible.

$$\frac{1}{2}-\frac{1}{3} p=\frac{4}{3}$$

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Solve each equation, if possible.

$$\frac{5}{8}(p+3)-2=\frac{1}{4}(2 p-3)+\frac{11}{16}$$

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Solve each equation, if possible.

$$\frac{1}{3}(w+1)-3=\frac{2}{5}(w-4)-\frac{2}{15}$$

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Solve each equation, if possible.

$$\frac{1}{2}+\frac{2}{x}=\frac{3}{4}$$

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Solve each equation, if possible.

$$\frac{3}{x}-\frac{1}{3}=\frac{1}{6}$$

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Solve each equation, if possible.

$$\frac{2 x}{x^{2}-4}=\frac{4}{x^{2}-4}-\frac{3}{x+2}$$

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Solve each equation, if possible.

$$\frac{x}{x^{2}-9}+\frac{4}{x+3}=\frac{3}{x^{2}-9}$$

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Solve each equation, if possible.

$$\frac{6 t+7}{4 t-1}=\frac{3 t+8}{2 t-4}$$

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Solve each equation, if possible.

$$\frac{8 w+5}{10 w-7}=\frac{4 w-3}{5 w+7}$$

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Solve each equation, if possible.

$$\frac{4}{x-2}=\frac{-3}{x+5}+\frac{7}{(x+5)(x-2)}$$

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Solve each equation, if possible.

$$\frac{-4}{2 x+3}+\frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)}$$

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Solve each equation, if possible.

$$\frac{2}{y+3}+\frac{3}{y-4}=\frac{5}{y+6}$$

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Solve each equation, if possible.

$$\frac{5}{5 z-11}+\frac{4}{2 z-3}=\frac{-3}{5-z}$$

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Solve each equation, if possible.

$$\frac{x}{x^{2}-9}-\frac{x-4}{x^{2}+3 x}=\frac{10}{x^{2}-3 x}$$

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Solve each equation, if possible.

$$\frac{x+1}{x^{2}+2 x}-\frac{x+4}{x^{2}+x}=\frac{-3}{x^{2}+3 x+2}$$

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Use a calculator to solve each equation. Round the solution to two decimal places.

$$3.2 x+\frac{21.3}{65.871}=19.23$$

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Use a calculator to solve each equation. Round the solution to two decimal places.

$$6.2 x-\frac{19.1}{83.72}=0.195$$

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Use a calculator to solve each equation. Round the solution to two decimal places.

$$14.72-21.58 x=\frac{18}{2.11} x+2.4$$

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Use a calculator to solve each equation. Round the solution to two decimal places.

$$18.63 x-\frac{21.2}{2.6}=\frac{14}{2.32} x-20$$

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Solve each equation. The letters $a, b,$ and $c$ are constants.

$a x-b=c, \quad a \neq 0$

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Solve each equation. The letters $a, b,$ and $c$ are constants.

$1-a x=b, \quad a \neq 0$

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Solve each equation. The letters $a, b,$ and $c$ are constants.

$\frac{x}{a}+\frac{x}{b}=c, a \neq 0, b \neq 0, a \neq-b$

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Solve each equation. The letters $a, b,$ and $c$ are constants.

$\frac{a}{x}+\frac{b}{x}=c, \quad c \neq 0$

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Find the number $a$ for which $x=4$ is a solution of the equation.

$$x+2 a=16+a x-6 a$$

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Find the number $b$ for which $x=2$ is a solution of the equation.

$$x+2 b=x-4+2 b x$$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Electricity $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ for $R$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Finance $A=P(1+r t)$ for $r$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Mechanics $F=\frac{m v^{2}}{R}$ for $R$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Chemistry $P V=n R T$ for $T$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Mathematics $S=\frac{a}{1-r}$ for $r$

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List some formulas that occur in applications. Solve each formula for the indicated variable.

Mechanics $v=-g t+v_{0} \quad$ for $t$

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A total of $\$ 20,000$ is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by $\$ 3000$, how much will be invested in each type of investment?

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A total of $\$ 10,000$ is to be divided between Sean and George, with George to receive $\$ 3000$ less than Sean. How much will each receive?

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Kim is paid time-and-a-half for hours worked in excess of 40 hours and had gross weekly wages of $\$ 910$ for 48 hours worked. What is her regular hourly rate?

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Leigh is paid time-and-a-half for hours worked in excess of 40 hours and double-time for hours worked on Sunday. If Leigh had gross weekly wages of $\$ 1083$ for working 50 hours, 4 of which were on Sunday, what is her regular hourly rate?

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Going into the final exam, which will count as two tests, Brooke has test scores of 80,83,71,61 , and $95 .$ What score does Brooke need on the final in order to have an average score of $80 ?$

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Going into the final exam, which will count as two-thirds of the final grade, Mike has test scores of $86,80,84,$ and $90 .$ What minimum score does Mike need on the final in order to earn a B, which requires an average score of $80 ?$ What does he need to earn an $A$, which requires an average of $90 ?$

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A store sells refurbished iPhones that cost $12 \%$ less than the original price. If the new price of a refurbished iPhone is $\$ 572,$ what was the original price? How much is saved by purchasing the refurbished phone?

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A car dealer, at a year-end clearance, reduces the list price of last year's models by $15 \%$. If a certain four-door model has a discounted price of $\$ 18,000,$ what was its list price? How much can be saved by purchasing last year's model?

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A movie theater marks up the candy it sells by $275 \%$. If a box of candy sells for $\$ 4.50$ at the theater, how much did the theater pay for the box?

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The suggested list price of a new car is $\$ 24,000$. The dealer's cost is $85 \%$ of list. How much will you pay if the dealer is willing to accept $\$ 300$ over cost for the car?

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The manager of the Coral Theater wants to know whether the majority of its patrons are adults or children. One day in July, 5200 tickets were sold and the receipts totaled $\$ 29,961 .$ The adult admission is $\$ 7.50$, and the children's admission is $\$ 4.50 .$ How many adult patrons were there?

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A pair of leather boots, discounted by $30 \%$ for a clearance sale, has a price tag of $\$ 399 .$ What was the original price?

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The perimeter of a rectangle is 60 feet. Find its length and width if the length is 8 feet longer than the width.

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The perimeter of a rectangle is 42 meters. Find its length and width if the length is twice the width.

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Herschel uses an app on his smartphone to keep track of his daily calories from meals. One day his calories from breakfast were 125 more than his calories from lunch, and his calories from dinner were 300 less than twice his calories from lunch. If his total caloric intake from meals was $2025,$ determine his calories for each meal.

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Tyshira tracks her net calories (calories taken in minus calories burned) as part of her fitness program. For one particular day, her net intake was 1480 calories. Her lunch calories were half her breakfast calories, and her dinner calories were 200 more than her breakfast calories. She ate 120 less calories in snacks than for breakfast, and she burned 700 calories by exercising on her elliptical. How many calories did she take in from snacks?

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Judy and Tom agree to share the cost of an $\$ 18$ pizza based on how much each ate. If Tom ate $\frac{2}{3}$ the amount that Judy ate, how much should each pay? [Hint: Some pizza may be left.]

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Find the largest perimeter of an isosceles triangle whose sides are of lengths $4 x+10,2 x+40$, and $3 x+18$.

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Solve:

$\frac{3}{4} x-\frac{1}{5}\left(\frac{1}{2}-3 x\right)+1=\frac{1}{4}\left(\frac{1}{20} x+6\right)-\frac{4}{5}$

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A regular hexagon is inscribed in a circle. Find the radius of the circle if the perimeter of the hexagon is 10 inches more than the radius.

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One step in the following list contains an error. Identify it and explain what is wrong.

$$\begin{aligned}x &=2 \quad (1) \\3 x-2 x &=2 \quad (2) \\3 x &=2 x+2 \quad (3) \\x^{2}+3 x &=x^{2}+2 x+2\quad (4) \\x^{2}+3 x-10 &=x^{2}+2 x-8\quad (5) \\(x-2)(x+5) &(x2(x+4)\quad(6)\\x+5 &=x+4\quad(7) \\1 &=0\quad(8)\end{aligned}$$

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The equation

$$\frac{5}{x+3}+3=\frac{8+x}{x+3}$$

has no solution, yet when we go through the process of solving it, we obtain $x=-3 .$ Write a brief paragraph to explain what causes this to happen.

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Make up an equation that has no solution and give it to a fellow student to solve. Ask the fellow student to write a critique of your equation.

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