Exercises: Algebraic language

Express the following situations related to saving and social networks in algebraic language:

1. a) The number of followers an influencer has, $x$, increased by $150$.
2. b) The price of a movie ticket, $p$, reduced by $\$2$.
3. c) Three times the age of my brother, if his age is $y$.
4. d) Half the money I have in my piggy bank, where $d$ is the total amount.

Copy and complete the following table identifying the components of these monomials:

Monomial	Coefficient	Variable part	Degree
$7x^2$
$-ab^3$
$\frac{1}{2}x^3y$
$-5$

From the following list of monomials, identify and group those that are like terms:

$5x^2$, $3y$, $-2x$, $\frac{1}{4}x^2$, $8y$, $x^2$, $-5y$, $10x$

Calculate the numerical value of the following expressions for the given values:

1. a) $3x + 10$ for $x = 4$
2. b) $x^2 - 5$ for $x = 3$
3. c) $2(a + b)$ for $a = 5$ and $b = 2$
4. d) $\frac{12}{x}$ for $x = 6$

Find the algebraic expression that represents the perimeter of the following square whose side measures $x$ feet:

Match each phrase with its corresponding algebraic expression:

1. Twice a number plus five units.                      a) $x - 5$

2. A number decreased by five units.                         b) $2x + 5$

3. A fifth of a number.                                           c) $x^5$

4. A number raised to the fifth power.                           d) $\frac{x}{5}$

In a video game shop, the cost of renting a console is a fixed $\$10$ plus $\$2$ for each hour of use.

1. a) Write the algebraic expression for the total cost for a number of hours $h$.
2. b) How much will Ashley pay if she rents the console for $3$ hours?

Find the algebraic expression that represents the area of the following rectangle, knowing that its base measures $5$ units more than its height ($x$):

On a video game streaming platform, the basic monthly fee is $\$12$. In addition, every time you want to play a "Premium" title outside the catalog, you must pay an additional $\$3,50$.

Write the algebraic expression that represents the total cost a user will pay per month if they decide to play $x$ "Premium" titles.

Calculate the algebraic expression for the perimeter of the following figure, which represents the plan of a small rectangular play area.

Observe the following sequence of figures formed by dots and determine the general term $a_n$ that allows you to calculate the number of dots needed for any figure $n$.

A recycling center awards "eco points" according to the material handed in:

• $5$ points for each pound of glass ($v$).

• $3$ points for each pound of paper ($p$).

• $10$ fixed points for each visit to the center.

Write the formula for the total points $P$ that a person receives in one visit. Once written, calculate how many points a student will receive who hands in $4$ pounds of glass and $8$ pounds of paper.

Analyze the following algebraic expressions and complete the information requested in each case:

1. In the expression $7x^2y - 5x + 3$: How many terms does it have? What is the constant term?

2. Given the monomial $-\frac{2}{3}ab^4$: State its coefficient, its variable part and its total degree.

Ashley is analyzing the growth of her followers on a social network. She started the month with $x$ followers. During the first two weeks, her number of followers tripled. However, in the second two weeks, she lost $150$ followers due to a cleanup of inactive accounts.

Write the algebraic expression that represents Ashley's final number of followers and calculate how many she has if she started the month with $x = 420$.

Translate these phrases from common language into algebraic language using a single variable $x$:

1. a) The sum of a number and its half.
2. b) Three times the result of subtracting five from a number.
3. c) The square of the previous number.
4. d) The difference between twice a number and the next number.

A cardboard shipping box has a square base of side $x$ and a height that is $5$ in. greater than the side of the base.

Draw a diagram of the rectangular prism and write the simplified algebraic expression that represents the volume of the box in terms of $x$.

A city council wants to install a wooden fence around a new playground. The fence is built by joining rectangular panels using vertical posts. Each panel needs $3$ horizontal slats. To support the structure, a post is placed at the beginning of the fence and another at the end of each panel.

1. Write an algebraic expression that allows calculating the total number of horizontal slats needed for a fence that has $n$ panels.

2. Write an algebraic expression for the total number of posts needed for a fence of $n$ panels.

3. If each post costs $\$p$ and each horizontal slat costs $\$l$, write the expression for the total cost of the materials for a fence of $n$ panels.

A courier company charges its customers according to the weight of the packages and the delivery distance. The base fare is $\$B$. To this amount is added:

• $\$2,5$ for each pound of weight ($w$).

• $\$0,15$ for each mile of distance ($d$).

If the customer is "Premium", a $15\%$ discount is applied to the total price of the delivery. Write the algebraic expression that determines the total cost of sending a package for a "Premium" customer as a function of $B$, $w$ and $d$.

From a square steel sheet with side $x$ inches, an operator must cut out four equal circles, one in each corner, to manufacture an industrial part. The radius of each circle is $r$ inches.

1. Write the algebraic expression for the area of the sheet that is left over after making the four cuts (use $\pi$ as a symbol).

2. If we know that the radius of the circles is a quarter of the side of the sheet, that is $r = \frac{x}{4}$, simplify the previous expression so that it depends solely on the variable $x$.

Observe the following sequence of figures formed by small hexagons. The first figure has a central hexagon surrounded by six others. In each step, a new layer of hexagons is added around the previous figure.

1. Determine how many hexagons layer number $n$ has (without counting those in the inner layers).

2. Find the algebraic expression that represents the total number of hexagons that make up the complete figure when it has $n$ layers around the central hexagon.

(Hint: The number of hexagons added in each layer follows a progression).

Translate the following properties relating to integers into algebraic language:

1. The square of the sum of any two numbers, $a$ and $b$, is equal to the sum of their squares plus twice their product.

2. The sum of the squares of three consecutive numbers, where $n$ is the smallest of them.

3. The difference between the cube of a number $x$ and the square of its half.

4. The quotient of the sum of a number and its reciprocal, and three times that number.

Three siblings, Mark, Ashley, and Ethan, decide to save up to buy a present. We know that:

• Ashley has saved $\$x$.

• Mark has saved $\$10$ less than three times what Ashley has.

• Ethan has saved half the sum of what his other two siblings have together.

Express in terms of $x$:

1. The money Mark has.

2. The money Ethan has (simplify the expression).

3. The total money saved by the three siblings.

We want to design a cardboard box without a lid from a rectangular sheet of cardstock measuring $50$ in. long by $30$ in. wide. To do this, a square of side $x$ is cut out from each of the corners and the flaps are folded up.

Write the simplified algebraic expression that represents:

1. The area of the base of the box in terms of $x$.

2. The total volume of the box in terms of $x$.

A streaming platform offers two types of subscription:

• Basic Plan: A fixed fee of $\$x$ per month plus $\$0,50$ for each new release movie rented ($p$).

• Premium Plan: A fixed fee that costs $\$40\%$ more than the basic one, but allows you to watch an unlimited number of movies at no extra cost.

• Write the algebraic expression for the monthly cost of the Basic Plan.

• Write the algebraic expression for the monthly cost of the Premium Plan.

• Write the expression that represents the difference in cost between the Premium Plan and the Basic Plan in a month where $p$ movies are rented. For what values of $p$ would the Basic Plan be more expensive than the Premium Plan? (State only the condition).