🧮 algebra

1. **State the problem:** We have a table with some values for Nuts and Raisins, and we want to use equivalent ratios to complete the missing values. 2. **Understand equivalent rat

1. The problem is to understand and use the quadratic formula to find the roots of a quadratic equation $ax^2 + bx + c = 0$. 2. The quadratic formula is given by:

1. **State the problem:** Simplify the expression $$\left(\frac{9x^{11}}{3x^3}\right)^{-4}$$. 2. **Use the quotient rule for exponents:** When dividing like bases, subtract the exp

1. **State the problem:** We are given a table showing the weight in pounds and the corresponding price in dollars. We need to find the constant of proportionality, which is the pr

1. **State the problem:** We need to solve the equation $$\frac{2x+4}{x-3} = 3$$ for $x$. 2. **Recall the formula and rules:** To solve a rational equation like this, multiply both

1. The problem asks to identify ratios equivalent to 15:12. 2. Two ratios are equivalent if their fractions are equal when simplified.

1. **Problem statement:** Find $(f \circ g)(x)$ and its domain, then find $(g \circ f)(x)$ and its domain for the functions given.

1. **State the problem:** Simplify the expression $$\frac{(4a^1 b^{-1})^{-2}}{4ab^{-4} c^0}$$ and verify if it equals $$\frac{1}{b^6}$$. 2. **Recall the rules:**

1. The problem asks to identify which ratios are equivalent to $11:16$. 2. Two ratios $a:b$ and $c:d$ are equivalent if their cross products are equal, i.e., $a \times d = b \times

1. **State the problem:** We need to divide the fraction $\frac{5}{6}$ by the fraction $\frac{4}{3}$. This is written as $\frac{5}{6} \div \frac{4}{3}$.\n\n2. **Recall the rule for

1. **State the problem:** We are given a system of inequalities:

1. The problem asks us to determine if the inequality $x < 6$ is true or false given that the elevations $x$ are all above 6 meters. 2. The inequality $x < 6$ means the elevation $

1. The problem is to understand and represent the inequality $w \geq -1 \frac{1}{2}$, which means $w$ is greater than or equal to $-1.5$. 2. The inequality $w \geq -1.5$ includes a

1. The problem asks to write an inequality representing "w is greater than or equal to -1 1/2". 2. The phrase "greater than or equal to" corresponds to the inequality symbol $\geq$

1. The problem is to understand the inequality $y < -8$ and its graphical representation on a number line. 2. The inequality $y < -8$ means that $y$ takes values less than $-8$, bu

1. The problem states: Write an inequality that represents the statement "y is less than -8" and identify its graph. 2. The inequality for "y is less than -8" is written as:

1. The problem asks to write an inequality representing the statement "y is less than -8". 2. The inequality symbol for "less than" is $<$.

1. The question asks about the origin of the term $7x$ in step 4. 2. Typically, $7x$ appears from combining like terms or distributing a coefficient in an algebraic expression.

1. **State the problem:** Solve the system of linear equations using substitution: $$\begin{cases} 2x + y = 5 \\ x - 3y = 13 \end{cases}$$

1. **State the problem:** Multiply and simplify the expression $2\sqrt{6} ( 5\sqrt{6} - 4\sqrt{2} )$. 2. **Recall the distributive property:** $a(b - c) = ab - ac$.

1. **State the problem:** We need to find a three-digit year where the digits satisfy the following conditions: - The tens digit is 1 less than the units digit.