🧮 algebra

1. **Problem:** Solve the inequality $$4 - x \leq 10$$ and graph the solution. 2. **Formula and rules:** To solve inequalities, isolate the variable on one side. Remember, when mul

1. **Stating the problem:** We have the line equation $$3x - 2y = a$$ and it cuts the y-axis at point $$P(0,4)$$. We need to find the constant $$a$$ and then find the x-intercept p

1. **State the problem:** We analyze the piecewise linear graph with points (-5, -2), (-3, 3), (-1, 3), (2, 5), (4, 1), and (6, 3) to identify intervals of increase, decrease, cons

1. **State the problem:** We are given a piecewise linear graph with points at (-5, -3), (-3, 1), (-1, 1), (1, 3), (3, 4), (4, 0), and (6, 2). We need to identify intervals where t

1. **State the problem:** Solve the quadratic equation $$8x^2 - 19x + 6 = 0$$ for $x$. 2. **Formula used:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, use the

1. **State the problem:** We need to find the pre-tax cost of the jeans given the final cost including tax is 37.82 and the tax rate is 6.5%. 2. **Formula used:** The final cost in

1. **State the problem:** Solve the system of equations: $$\sqrt{x} + 2y = 2$$

1. **State the problem:** We need to find the original bill amount before the 12% tip was added, given the final bill is 51.28. 2. **Formula used:** If the original bill is $x$, th

1. **State the problem:** Factorize the expression $$x^2 + 5x + 6x^2 + 5x + 6$$. 2. **Combine like terms:** Group the terms with the same powers of $x$.

1. **Problem 11:** Solve $$1 = \frac{v + 2}{v - 4} + \frac{7v - 42}{v - 4}$$ Since denominators are the same, combine numerators:

1. Solve the equation $$\frac{1}{v} + \frac{3v + 12}{v^2 - 5v} = \frac{7v - 56}{v^2 - 5v}$$ - First, factor the denominators: $$v^2 - 5v = v(v - 5)$$.

1. **Problem Statement:** Factorize the expression $$m^2 - 6m - 18mk + 108k$$. 2. **Identify the terms:** The expression has four terms: $$m^2$$, $$-6m$$, $$-18mk$$, and $$108k$$.

1. **Problem Statement:** Determine the range of the function given the points at $x=1$ with $y$-values 2, 3, 4, and 5. 2. **Understanding the Range:** The range of a function is t

1. **Stating the problem:** Factorize the expression $$5a^6 b^2 - 15a^3 b^5 + 35a^4 b^4$$. 2. **Identify the greatest common factor (GCF):**

1. The problem asks us to determine the range of the function given the points on its graph. 2. The range of a function is the set of all possible output values (y-values).

1. **State the problem:** We want to find the number of cookies $x$ Nancy must sell to maximize her monthly profit given by the function $$P(x) = 50\sqrt{x} - 0.5x - 75.$$\n\n2. **

1. **State the problem:** We need to find the selling price of a single book so that the publisher earns a 22% profit, given that 1,250 copies are published and shipping costs R52,

1. **State the problem:** Factorize the expression $$x^2 + 5x + 6x^2 + 5x + 6$$. 2. **Combine like terms:** Group the terms with the same powers of $x$.

1. **State the problem:** Solve the inequality $$-1 \leq 1 - \frac{4t}{5}$$ for $t$. 2. **Write down the inequality:** $$-1 \leq 1 - \frac{4t}{5}$$

1. **State the problem:** Solve the inequality $$-1 \leq 1 - \frac{4t}{5}$$. 2. **Isolate the variable term:** Subtract 1 from both sides:

1. The problem asks to find the antilogarithm of 2.5620. 2. The antilogarithm (or inverse logarithm) of a number $x$ is given by $10^x$ when the logarithm base is 10.