🧮 algebra

1. **State the problem:** We need to find the value of the piecewise function $f(x)$ at $x = -6$. 2. **Recall the piecewise definition:**

1. **Stating the problem:** Solve the equation $$(x - 1)^2 = -4$$ for $x$. 2. **Understanding the equation:** The left side is a square of a real number $(x-1)$, which means it is

1. **State the problem:** We have two bread recipes, each with a different rate of flour used per loaf. We want to find which recipe uses less flour per loaf and how much flour the

1. **State the problem:** We need to determine who types more words per minute between Mitch and Carrie, and then find how many words that person can type in 5 minutes. 2. **Given

1. **Problem:** Find the Least Common Multiple (LCM) of the numbers 16, 2, and 45. 2. **Formula and Explanation:** The LCM of a set of numbers is the smallest positive integer that

1. **State the problem:** We need to determine which machine produces juice boxes at a faster rate and then find how many juice boxes that faster machine produces in 30 minutes. 2.

1. **State the problem:** We need to compare the printing rates of two printer models: a solid ink printer and a laser printer, to determine which prints faster and how many pages

1. **State the problem:** We need to find who earns more per hour between Ashley and Kendra, and by how much more. 2. **Ashley’s hourly rate:** From the table, Ashley’s earnings ar

1. **State the problem:** We need to find who earns more per hour between Ashley and Kendra and by how much more one earns per hour. 2. **Find Ashley's hourly rate:** From Ashley's

1. The problem asks for the rate of decay of the function $$f(x) = 500 \left(\frac{1}{4}\right)^x$$. 2. The general form of an exponential decay function is $$f(x) = a(1 - r)^x$$ w

1. The problem states that the function $f(x) = 1.69(1.03)^x$ models the value of an investment after $x$ years, where $1.69$ is the initial amount (in thousands) and $1.03$ is the

1. Problem: Dane są funkcje $f(x) = \log_{\frac{1}{3}}(x - 1)$, $g(x) = \log_{\frac{1}{3}}(x + 2)$, $h(x) = \log_3 x + 2$, $k(x) = \log_3 x - 1$. Znajdź, które dwie funkcje przecho

1. **State the problem:** Solve the equation $$x^6 + 28x^3 + 27 = 0$$ for $x$. 2. **Rewrite the equation:** Let $$y = x^3$$, then the equation becomes $$y^2 + 28y + 27 = 0$$.

1. **State the problem:** Simplify the expression $4(12+3p)$. 2. **Formula used:** Use the distributive property of multiplication over addition: $a(b+c) = ab + ac$.

1. **State the problem:** Simplify the expression $3x + 15$ to a single number. 2. **Understand the expression:** The expression $3x + 15$ depends on the variable $x$. To get a sin

1. **State the problem:** We have a blueprint where Fence A is represented as 1/5 feet in real life but measures 1 4/5 inches on the blueprint. We need to find the unit rate in inc

1. **State the problem:** Simplify the expression $3x + 15$. 2. **Identify the formula and rules:** To simplify expressions like this, we look for common factors in each term.

1. **Stating the problem:** We have two fences with lengths $\frac{1}{4}$ feet and $\frac{1}{2}$ inches. We want to find the unit rate in inches per foot and then find the length o

1. **State the problem:** We need to find the unit rate in pounds per gallon given the quantities: $\frac{1}{6}$ pound, $\frac{1}{108}$ pound per gallon, and $\frac{1}{18}$ gallon.

1. **State the problem:** Jessie said the rate 1/8 gallon can be written as the unit rate 1/32 pound per gallon, but this is incorrect. We need to find the correct unit rate in pou

1. **Stating the problem:** We are given several unit rates and fractions involving pounds and quarts: - 1/4 pound