🧮 algebra

1. **State the problem:** Solve the equation $$\frac{r}{4} + 9 = 21$$ for the variable $r$. 2. **Isolate the term with $r$:** To isolate $\frac{r}{4}$, subtract 9 from both sides:

1. The problem is to solve the equation $$\frac{x - 5}{4} = 3$$ for $x$. 2. The formula used here is to isolate $x$ by undoing the operations step-by-step. Since $x$ is subtracted

1. **State the problem:** Expand and fully simplify the expression $$(x^2 + 3x + 2)(x + 8)$$. 2. **Formula and rules:** To expand, use the distributive property (also called FOIL f

1. **State the problem:** Solve the quadratic expression $w^2 - 15w + 54 = 0$ for $w$. 2. **Formula and rules:** To solve a quadratic equation $ax^2 + bx + c = 0$, we can factor it

1. **State the problem:** Fully factorise the quadratic expression $x^2 - 20x - 44$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers t

1. **Stating the problem:** We are given an arithmetic sequence defined by terms involving $P$ and $E$, specifically the terms $P$, $E$, $1000E$, and $-P$. We need to analyze or fi

1. Énoncé du problème : Calculer les images des réels donnés par les fonctions $f$ et $g$ définies par $$f(x) = \frac{1}{x^2 + 2}$$

1. The problem states that 40 more students like vanilla ice cream than those who do not. 2. From the pie chart, the proportion of students who like vanilla ice cream is represente

1. **State the problem:** We need to find the solutions to the equation $$-\frac{1}{2} x^2 - x + 8 = x + 3$$ which represent the points where the parabola and the line intersect. 2

1. **State the problem:** We are given the function $$f(x) = -2x^2 + 10x + 1$$ and the set $$X = \{1, 3, 5, 7, 9\}$$. We need to find the maximum value of $$f(x)$$ for $$x \in X$$.

1. The problem is to evaluate $46^0$. 2. The rule for any nonzero number raised to the power of zero is:

1. We are given two functions: $f(x) = \log_3(2x+1)$ and $g(f(x)) = 4x^2$. We need to find the function $y = g(x)$.\n\n2. The problem states that $g$ is composed with $f$, so $g(f(

1. The problem states that we have two lines: a vertical line at $x=3$ and a horizontal line at $y=-2$. 2. A vertical line at $x=3$ means all points on this line have an $x$-coordi

1. **State the problem:** Solve the equation $$\ln(x + 70) + \ln x = \ln 71$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms is the logarithm of the product:

1. **Problem 1:** Solve the equation $|x+1| = e$. - The absolute value equation $|x+1| = e$ means $x+1 = e$ or $x+1 = -e$.

1. **Problem 1:** Solve the equation $$\ln(x + 70) + \ln x = \ln 71$$. 2. **Problem 2:** Solve the equation $$2 \cdot 6^{m+2} + 4 = 44$$.

1. **State the problem:** Sidney used 6241 units of electricity. The first 2400 units cost 7p each, and all units beyond 2400 cost 12p each. We need to find the total cost.

1. **State the problem:** Solve the equation $$\ln(x - 8) + \ln 3 = 3$$ given the square root values 4 and 25 (which might be hints for simplification). 2. **Recall the logarithm p

1. **State the problem:** Given the equation $\log_3 x + \log_3 y = 2$, find the minimum value of $2x + 3y$. 2. **Use logarithm properties:** Recall that $\log_a b + \log_a c = \lo

1. **State the problem:** Solve the equation $\ln 2 - \ln (x + 1) = 3$ for $x$. 2. **Recall the logarithm property:** The difference of logarithms can be written as the logarithm o

1. **State the problem:** Solve the equations \( \log(x(x+21)^{10}) = 2 \) and \( \ln 2 - \ln (x+1) = 3 \). 2. **Recall logarithm properties:**