🧮 algebra

1. Problem: Determine if the given ratios are true or false. 2. For part a: The ratio 3 : 2 is compared to 2 : 3.

1. **State the problem:** Solve the proportion $$\frac{30}{40} = \frac{18}{x+18}$$ for $x$. 2. **Use the cross-multiplication formula for proportions:** If $$\frac{a}{b} = \frac{c}

1. The problem states that the variable $w$ represents the number 2. 2. This means wherever you see $w$ in an expression or equation, you substitute it with 2.

1. **Problem statement:** Santa wants to build a rectangular enclosure using fencing and wants to find the dimensions that maximize the area of the enclosure. 2. **Formula and rule

1. **State the problem:** Simplify the expression $\frac{21w}{10} - w - 1$. 2. **Recall the rule:** To combine terms with the variable $w$, they must have a common denominator.

1. **Problem:** Compare the fractions $\frac{3}{4}$ and $\frac{5}{8}$. 2. **Formula and rule:** To compare fractions with different denominators, use cross multiplication:

1. **State the problem:** Solve the proportion $$\frac{3}{14} = \frac{15}{m-3}$$ for $m$. 2. **Use the cross-multiplication formula:** For proportions $\frac{a}{b} = \frac{c}{d}$,

1. **State the problem:** We need to select all the squares from the given expressions and order them from least to greatest. 2. **Identify squares:** A square is a number raised t

1. **Problem:** Identify the greatest common factor (GCF) for each expression. 2. **Formula:** The GCF is the largest factor that divides all terms in the polynomial.

1. **Problem:** Solve the system of equations $$5x - 19 = y$$

1. **State the problem:** Solve the inequality $5x > 15$. 2. **Formula and rules:** To solve inequalities, we isolate the variable on one side. When dividing or multiplying both si

1. The problem is to simplify the expression $5|x|^{10}\pi$. 2. The expression involves the absolute value of $x$ raised to the 10th power, multiplied by 5 and $\pi$.

1. **State the problem:** Find the coordinates of the minimum point of the curve given by the function $$y = 2x^2(3 + x)$$. 2. **Rewrite the function:** Expand the expression to si

1. **Problem statement:** Familien Riveras har et trækar med ydre dimensioner: længde 180 cm, bredde 80 cm, højde 65 cm, og træpladernes tykkelse er 7 cm.

1. Дано уравнение: $a^2 = 2a \times a + b$. 2. Раскроем произведение в правой части: $2a \times a = 2a^2$.

1. **State the problem:** Solve the equation $a^2 = 2a \times a + b$ for $b$. 2. **Rewrite the equation:** The equation is given as

1. **State the problem:** Solve the cubic equation $$2x^3 - 3x^2 - 9x + 10 = 0$$ for $x$. 2. **Recall the approach:** For cubic equations, we can try to find rational roots using t

1. **State the problem:** Simplify the expressions: a) $\frac{6^n + 3^n}{2^{n+1} + 2}$

1. Simplify the expression $\frac{x}{2} + \frac{3x}{4}$. 2. To add fractions, find a common denominator. Here, denominators are 2 and 4. The least common denominator (LCD) is 4.

1. **State the problem:** We are given four points representing average heart rate (y) at different running speeds (x): (8,140), (10,160), (12,180), and (14,200). We want to find t

1. **State the problem:** Find the equation of the straight line passing through the point $(4,14)$ with gradient $2$. 2. **Recall the formula:** The equation of a straight line is