Solving Equations and Inequalities

Name the property illustrate by each statement.

If $\left[3(-2)\right]z=24$, then $-6z=24$.

Name the property illustrate by each statement.

If $\mathbf{5}\mathbf{+}\mathit{b}\mathbf{=}\mathbf{13}$, then $b=8$.

Name the property illustrate by each statement.

If $2x=3d$and $3d=-4$, then $2x=-4$.

Name the property illustrate by each statement.

If $g-t=n$, then $g=n+t$.

Name the property illustrate by each statement.

If $14=\frac{x}{2}+11$, then $\frac{x}{2}+11=14$.

Name the property illustrate by each statement.

If $y-2=-8$, then $3\left(y-2\right)=3\left(-8\right)$.

Solve each equation. Check your solution.

$2p+15=29$

Solve each equation. Check your solution.

 $14-3n=-10$

Solve each equation. Check your solution.

$7a-3a+2a-a=16$

Solve each equation. Check your solution.

$x+9x-6x+4x=20$

Solve each equation. Check your solution.

$\frac{1}{9}-\frac{2}{3}b=\frac{1}{18}$

Solve each equation. Check your solution.

$\frac{5}{8}+\frac{3}{4}x=\frac{1}{16}$

Solve each equation. Check your solution.

$27=-9\left(y+5\right)$

Solve each equation. Check your solution. 

$-7\left(p+8\right)=21$

Solve each equation. Check your solution.

$3f-2=4f+5$

Solve each equation. Check your solution.

$3d+7=6d+5$

Solve each equation. Check your solution.

$4.3n+1=7-1.7n$

Solve each equation. Check your solution.

$1.7x-8=2.7x+4$

Solve each equation. Check your solution.

$3\left(2z+25\right)-2\left(z-1\right)=78$

Solve each equation. Check your solution.

$4\left(k+3\right)+2=4.5\left(k+1\right)$

Solve each equation. Check your solution.

$\frac{3}{11}a-1=\frac{7}{11}a+9$

Solve each equation. Check your solution.

$\frac{2}{5}x+\frac{3}{7}=1-\frac{4}{7}x$

Solve each equation or formula for the specified variable.

$d=rt$, for r.

Solve each equation or formula for the specified variable.

$x=\frac{-b}{2a}$, for a.

Solve each equation or formula for the specified variable.

$V=\frac{1}{3}\pi r^{2}h$, for h.

Solve each equation or formula for the specified variable.

$A=\frac{1}{2}h\left(a+b\right)$, for b.

Solve each equation or formula for the specified variable.

$\frac{a\left(b-2\right)}{c-3}=x$, for b.

Solve each equation or formula for the specified variable. 

$x=\frac{y}{y+4}$, for y.

Define a variable, write an equation and solve the problem.

BOWLING Jon and Morgan arrive at Sunnybrook Lanes with $\backslash (16.75$. Find the maximum number of games they can bowl if they each rent shoes.

Shoe Rental: $\backslash )1.50$

Games: $\$2.50$

The perimeter of a regular octagon is 124 inches, Find the length of each side.

For the exercise 65, define a variable, write an equation, and solve the problem.

CAR EXPENSES  Benito spent $\backslash (1837$ to operate his car last year. Some of these expenses are listed below. Benito’s only other expense was for gasoline. If he drove 7600 miles, what was the average cost of the gasoline per mile?

Operating expenses

Insurance: $\backslash )972$

Registration: $\backslash (114$

Maintenance: $\backslash )105$

A school conference room can seat a maximum of 83 people. The principal and two counselors need to meet with the school’s student athletes to discuss eligibility requirements. If each student must bring a parent with them, what is the maximum number of students that can attend each meeting?

For the exercise 67, define a variable, write an equation, and solve the problem.

FAMILY  Chun-Wei’s mother is 8 more than twice his age. His father is three years older than his mother is. If all three family members have lived 94 years, how old is each family member?

For the exercise 68, define a variable, write an equation, and solve the problem.

The Parent Teacher Organization has raised $\backslash (1800$ to help pay for a trip to an amusement park. They ask that there be one adult for every five students attending. Adult tickets cost $\backslash )45$ and student ticket cost $\$30$. If the group wants to take 50 students, how much will each student need to pay so that adults agreeing to chaperone pay nothing?

For the exercise 69, define a variable, write an equation, and solve the problem.

BUSINESS A trucking company is hired to deliver 125 lamps for $\backslash (12$each. The company agrees to pay $\backslash )45$ for each lamp that is broken during transport. If the trucking company needs to receive a minimum payment of $\$1364$ for the shipment to cover their expenses, find the maximum number of lamps they can afford to break during the trip.

For Exercises 64-70, define a variable, write an equation, and solve the problem.

Two designs for a soup can are shown at the right. If each holds the same amount of soup, what is the height of can A?

For exercises 71-73, use the following information.

The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westwards from Omaha, Nebraska. The two lines met at Promontory, Utah, in 1869, about 6 years after construction began.

71. The Central Pacific Company laid an average of $\mathbf{9}\mathbf{.}\mathbf{6}$ miles of track per month. Together the two companies laid a total of 1775 miles of track. Determine the average number of miles of track laid per month by the Union Pacific company.  

For exercises 71-73, use the following information.

The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westwards from Omaha, Nebraska. The two lines met at Promontory, Utah, in 1869, about 6 years after construction began.

72. About how many miles of track did each company lay.

For exercises 71-73, use the following information.

The First Transcontinental Railroad was built by two companies. The Central Pacific began building eastward from Sacramento, California, while the Union Pacific built westwards from Omaha, Nebraska. The two lines met at Promontory, Utah, in 1869, about 6 years after construction began.

Why do you think the Union Pacific was able to lay track so much more quickly than the Central Pacific.

Allison is saving money to buy a video game system. In the first week, her savings were \(8 less than $\frac{\mathbf{2}}{\mathbf{5}}$ the price of the system. In the second week, she saved 50 cents more than $\frac{\mathbf{1}}{\mathbf{2}}$ the price of the system. She was still \)37 short. Find the price of the system.

Write a verbal expression to represent the algebraic expression $3\left(x-5\right)+4x\left(x+1\right)$.

Answer the question that was posed at the beginning of the lesson.

How can you find the most effective level of intensity for your workout?

Include the following in your answer:

• An explanation of how to find the age of a person who is exercising at an $80\%$ level of intensity I  with a pulse count of 27, and 
• A description of when it would be desirable to solve a formula like the one given for a specific variable.

If $-6x+10=17$, then $3x-5=$

A. $-\frac{7}{6}$

B. $-\frac{17}{2}$

C. 2

D. $\frac{19}{3}$

E. $\frac{5}{3}$

In triangle PQR, $\overline{QS}$ and $\overline{SR}$ are angle bisectors and angle $P=74°$. How many degrees are there in angle QSR?

A. 106

B. 121

C. 125

D. 127

E. 143

Simplify each expression.

$2x+9y+4z-y-8x$

Simplify each expression.

$4\left(2a+5b\right)-3\left(4b-a\right)$

Evaluate each expression if $a=3$, $b=-2$, and $c=1.2$.

$a-\left[b\left(a-c\right)\right]$

Evaluate each expression if $a=3$, $b=-2$, and $c=1.2$.

$c^2-ab$

The formula for the surface area S of a regular pyramid is $S=\frac{1}{2}Pl+B$, where P is the perimeter of the base, l is the slant height, and B is the area of the base. Find the surface area of the square based pyramid shown at the right.

Identify the additive inverse for each number or expression.

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