🧮 algebra

1. **Simplify the expression $\sqrt{26} - 1$.** Since $\sqrt{26}$ is approximately 5.099, the expression is approximately

1. **State the problem:** We want to describe and understand the region defined by the inequalities: $$-1 < x < 4$$

1. We are asked to add the two fractions $\frac{4}{10}$ and $\frac{52}{60}$. 2. To add fractions, we first find a common denominator. The denominators are 10 and 60.

1. The problem is to add the fractions $\frac{4}{10}$ and $5\ \frac{2}{20}$.\n2. Convert the mixed number $5\ \frac{2}{20}$ to an improper fraction: $5 = \frac{5 \times 20}{20} = \

1. We are asked to find the non-real solutions to the quadratic equation $x^2 + 4x + 5 = 0$ using the quadratic formula. 2. Recall the quadratic formula for solutions of $ax^2 + bx

1. The topic is analyzing algebraic functions, which involves understanding properties like intercepts, extrema, domain, and range. 2. A beginner-level question could be: Analyze t

1. **Problem statement:** Given sets $A = B = \{x \mid -2 \leq x \leq 2\}$ and functions: - (a) $f(x) = |x|$

1. Stating the problem: Solve the equation $$\frac{14}{2^x + 3} + \frac{15}{2^x + 1} = 5$$ for $x$. 2. Let $y = 2^x$. Since $2^x > 0$ for all real $x$, we have $y > 0$.

1. **State the problem:** Solve for $x$ in the equation $$\frac{14}{2^x}+3 + \frac{15}{2^x} + 1 = 5.$$\n\n2. **Combine like terms:** Group terms with $2^x$: $$\frac{14}{2^x} + \fra

1. **Énoncé du problème** : Démontrer que la relation $R$ sur $G$ définie par $xRy \iff x = y$ ou $x = y^{-1}$ est une relation d'équivalence.

1. We start with the equation: $$16^{1/x} - 20 \cdot 2^{(2/x)-2} + 4 = 0$$ 2. Rewrite 16 as a power of 2: $$16 = 2^4$$, so $$16^{1/x} = (2^4)^{1/x} = 2^{4/x}$$.

1. **State the problem:** Solve the system of linear equations by matrix methods for two systems: ①\

1. **Problem:** Suppose $R = \{(x,y): y \geq x^2 - 4 \text{ and } y < x + 2\}$. Find the domain and range of $R^{-1}$. **Step 1:** The relation $R$ consists of points $(x,y)$ where

1. Problem: Factor out the common binomial factor $(x+2)$ from an expression. 2. Suppose the expression is of the form $(x+2)(ax+b) + (x+2)(cx+d)$. Our goal is to factor $(x+2)$ ou

1. **State the problem:** We are given the function $f(x) = |x|$ and need to find its range. 2. **Recall the definition of the absolute value:** The absolute value $|x|$ of a numbe

1. The problem gives two functions: a) $f(x) = |x|$

1. \nWe are given three points: $(-3, a)$, $(0, 4)$, and $(6, 2a)$. We need to find $2a + 4$ given that these points are collinear.\n\n2. \nIf three points are collinear, the slope

1. **Énoncé du problème :** Étudier la fonction quadratique $$f(x) = (m+2)x^2 - 2(m+1)x + m - 1$$ avec $m \neq -2$. On demande de discuter les solutions de l'équation $f(x)=0$, ana

1. Stating the problem: Solve the equation $$2^{\frac{1}{2x}} + 2^{\frac{1}{x}} = 6$$ for $x$. 2. Introduce a substitution: Let $$y = 2^{\frac{1}{2x}}$$. Then, $$2^{\frac{1}{x}} =

1. **State the problem:** Solve the equation $$\frac{1}{2x} + \frac{1}{2} = 6$$ for $x$. 2. **Isolate the fraction terms:** Write the equation clearly:

1. **State the problem:** We need to simplify the function $f(x) = \frac{x^2 - 9}{x - 3}$ and identify which linear function among the options A, B, C, and D represents $f(x)$. 2.