🧮 algebra

1. **State the problem:** We are given a rectangle with sides \( \frac{2}{x-2} \) and \( \frac{3}{x-1} \), and the perimeter is 10 cm. We need to write an equation for \( x \) and

1. The problem asks us to find the restrictions on the variable $x$ so that the expression $\sqrt{3-2x}$ is defined. 2. Recall that the square root function $\sqrt{y}$ is defined o

1. **State the problem:** We are given the equation $$(e - 2\sqrt{3})^2 = f - 20\sqrt{3}$$ where $e$ and $f$ are integers. We need to find the values of $e$ and $f$.

1. **State the problem:** We want to expand and simplify the expression $$(a + \sqrt{8})^2$$ and write it in the form $$c + d\sqrt{2}$$ where $a$, $c$, and $d$ are integers. 2. **R

1. **State the problem:** We have the function $f(x,y) = x^2 e^{3xy}$. (a) Evaluate $f(1,1)$.

1. **State the problem:** Simplify the expression $-23 \pm \frac{\sqrt{-60}}{6}$. 2. **Recall the formula and rules:** The square root of a negative number involves imaginary numbe

1. **State the problem:** Simplify the expression $$\sqrt[3]{56x^7y^5}$$ where $$x \neq 0$$ and $$y \neq 0$$. 2. **Recall the cube root property:** For any positive integers $$a, b

1. **State the problem:** Solve the equation $$\sqrt{-2x - 5} - 4 = x$$ for $x$. 2. **Isolate the square root:** Add 4 to both sides:

1. **Stating the problem:** We will explore how to solve inequalities and apply consumer arithmetic concepts such as discounts, taxes, and budgeting. 2. **Inequalities basics:** An

1. **Stating the problem:** We are asked to solve an inequality problem related to consumer arithmetic. 2. **General approach:** Inequalities are solved similarly to equations but

1. We are asked to evaluate the expression $840 - 10^2 \times 6$. 2. According to the order of operations (PEMDAS/BODMAS), we first calculate exponents, then multiplication and div

1. **State the problem:** Solve the system of equations by substitution: $$y = 4x + 8$$

1. **Expand the brackets for the first expression:** Given: $2(6r - 7s + t)$

1. **State the problem:** Conor pays a fixed tax of 3440 plus 22% on the amount his taxable income exceeds 36000. His total tax is 3684. Find his taxable income.

1. Simplify the expressions: 1.a) Simplify $y^2 + 2y - 5y^2 - 1 - 2 + y$

1. The problem is to solve for $x$ in the equation shown in the image (assuming it is $2x + 3 = 7$). 2. The formula to isolate $x$ is to subtract 3 from both sides and then divide

1. **State the problem:** We need to solve the expression $$0.5 \times \frac{7}{6} - \frac{11}{8}$$. 2. **Recall the rules:** Multiplication and subtraction of fractions require a

1. **State the problem:** We need to solve the expression $$\frac{1}{6} + \frac{11}{15} \div \frac{5}{2}$$. 2. **Recall the order of operations:** Division and multiplication are p

1. State the problem: Simplify the expression $$\frac{\frac{1}{5}}{\frac{-8}{5}} + \frac{5}{6}$$. 2. Simplify the complex fraction by multiplying the numerator by the reciprocal of

1. Stating the problem: Add the fractions $-\frac{5}{8}$, $\frac{5}{1}$, and $\frac{6}{5}$.\n\n2. Find a common denominator: The denominators are 8, 1, and 5. The least common deno

1. **State the problem:** Evaluate the expression $$\frac{4}{3} \div \left( \frac{11}{12} - \frac{9}{8} \right)$$ and write the answer as a fraction or mixed number in simplest for