🧮 algebra

1. The problem states that the terms in the sequence increase by the same number each time, meaning it is an arithmetic sequence. 2. We are given Term 1 = 3 and Term 5 = 19.

1. The problem asks us to find which two recurring decimals among $0.3\dot{5}$, $0.35\dot{3}$, $0.35\dot{3}$, and $0.3\dot{5}$ are equivalent. 2. Let's write each recurring decimal

1. Let's analyze the behavior of the function as $x$ approaches negative infinity. 2. Suppose the function is $y = a^x$ where $a > 0$ and $a \neq 1$.

1. **State the problem:** We need to sketch a polynomial $P(x)$ with zeros at $-5$, $1$, and $4$, and with end behavior such that as $x \to \infty$, $P(x) \to -\infty$, and as $x \

1. **State the problem:** Calculate the square root of $$18 - 12\sqrt{2}$$. 2. **Rewrite the expression:** We want to find $$\sqrt{18 - 12\sqrt{2}}$$.

1. **Problem 1: Multiple Choice with Justifications** 1) Calculate $|\sqrt{5} - 2| + |2 - \sqrt{5}|$.

1. **State the problem:** Solve the equation $$|4^{y+1} - 1| = |4^y - 3|$$ correct to 3 significant figures. 2. **Rewrite the equation:** Note that $$4^{y+1} = 4 \cdot 4^y$$, so th

1. The problem is to simplify the expression $\sqrt{-x^2}$.\n\n2. Recall that $x^2$ is always non-negative for any real number $x$, so $-x^2$ is non-positive (less than or equal to

1. Problem 12: A car uses 5 liters of gasoline to travel 40 km. How much gasoline is needed to travel 128 km at the same rate? Step 1: Find the rate of gasoline consumption per km.

1. Justify that $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8})^{12} = 4\sqrt{14} - 15$$ Step 1: Recognize the conjugate terms and rewrite the expression.

1. **Justify that** $$(\sqrt{7} - \sqrt{8})^{11} (\sqrt{7} + \sqrt{8}) = 4 \sqrt{14} - 15$$ - Start by noting the conjugate multiplication: $$(\sqrt{7} - \sqrt{8})(\sqrt{7} + \sqrt

1. Given the system of equations: $$x + y - z = 2$$

1. **State the problem:** We are given two numbers, $x$ and $y$. The sum of twice the first number and the second number is 22, and subtracting the second number from thrice the fi

1. **Problem b)** Given the inequality $|c - 10| + 12 > 18$, find the interval of $c$. 2. Start by isolating the absolute value term:

1. The problem asks to analyze the algebraic expression $3x + 4$ and answer four questions about it. 2. First, count the number of terms in $3x + 4$. A term is a part of the expres

1. The problem is to factor the given algebraic expressions and identify equivalent expressions. 2. Factor each expression:

1. **State the problem:** We have two functions:

1. **State the problem:** We need to find the current ages of Muneer and Mujeeb given two conditions about their ages in the past and future. 2. **Define variables:** Let $M$ be Mu

1. **Factoring algebraic expressions:** Given: $2x + 10$

1. Let's define variables for the prices: - Let $C$ be the cost of one coat.

1. **Problem:** Find the value of $x$ such that \[ [x \quad 1] \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = [0 \quad 0] \] 2. **Step 1:** Perform the matrix multiplication: