🧮 algebra

1. Let's analyze the system of inequalities for each case and find the feasible regions. 2. For A:

1. Stating the problem: Solve the equation $$|X + 1| = 5$$. 2. Recall that absolute value equation $$|A| = B$$ (where $$B \geq 0$$) can be rewritten as two separate equations: $$A

1. **State the problem**: Given the cubic function $$F(x) = -\frac{1}{9}x^3 + 7x^2 - 7x - 15,$$ find its roots, critical points, and analyze sign changes. 2. **Find roots of the fu

1. Stating the problems: We are solving for $x$ in the absolute value equations:

1. The problem states we have two parabolic curves derived from birth months and years. 2. For the father: birth month is June (6), birth year ends with 20.

1. Simplify expression 3a + 6b by factoring out the common factor 3: $$3a + 6b = 3(a + 2b)$$

1. Sketch the graph of each function. **1) $y = 2x^2 + 6x$**

1. **State the problem:** Expand the expression $ (5x + 2y)(3x - 2y) $. 2. **Apply the distributive property (FOIL method):** Multiply each term in the first parenthesis by each te

1. **State the problem:** Solve the quadratic equation $$x^2 - 6x + 9 = 1$$ using square roots and check the solutions. 2. **Rewrite the equation:** Subtract 1 from both sides to s

1. **State the problem:** We want to find a whole number $x$ such that the sum of $x$ and twice the square of $x$ equals 10. 2. **Write an equation:** The problem translates to the

1. **Write the equation of a line parallel to** $3x - 2y + 5 = 0$ **and passing through** $(1,1)$. - Lines parallel have the same slope. First, rewrite the given line in slope-inte

1. The problem gives us two points: Laurie at $(-5,30)$ and Anne at $(23,9)$, and mentions a fountain halfway between them. 2. To find the fountain's coordinates, we calculate the

1. **State the problem:** Simplify the expression $$\sqrt{7 - t} + \sqrt{7 + t}$$ and determine when its value is a natural number (i.e., an integer greater than or equal to 1). 2.

1. Problem 7: If four times a whole number is subtracted from three times the square of the number, the result 15 is obtained. Find the number. Step 1: Let the whole number be $x$.

1. Stated problem: Factor the quadratic expression $x^2 + 5x + 6$. 2. To factor, find two numbers that multiply to $6$ (the constant term) and add to $5$ (the coefficient of $x$).

1. The problem asks to simplify the expression $$\sqrt{7} - t + \sqrt{7} + t$$ and determine under what conditions the result is a natural number (belongs to $\mathbb{N}$). 2. Begi

1. The problem is to describe the function $y = x^3$. 2. This is a cubic function where each input $x$ is raised to the third power.

1. The problem is to simplify or solve an expression where the terms $-t$ and $t$ are inside a square root. 2. Let the expression inside the root be $-t + t$ or $t - t$; inside the

1. First, state the problem: Prove that $\sqrt{7} - t + \sqrt{7} + t \in \mathbb{N}$ where $t$ is a variable. 2. Simplify the expression inside the set membership:

1. The problem is to solve an algebraic expression or equation. 2. Since no specific problem was given, let's consider a general example: Solve for $x$ in the equation $$2x + 3 = 1

1. We are asked to determine certain key features of the quadratic function $f(x)$. 2. The vertex form of a quadratic function $f(x) = ax^2 + bx + c$ is given by the vertex coordin