🧮 algebra

1. Simplify the expression \(\frac{5x+5}{3(2x-1)} - \frac{6-2x}{2(1-2x)}\). 2. Note that \(1-2x = -(2x-1)\), so rewrite the second denominator:

1. **Problem statement:** Simplify the expressions: (a) $\frac{4a + 13}{5} - \frac{2a + 3}{3}$

1. Stating the problem: Simplify the expression $$\frac{-3a^2 \times 4a^{-1} b^{-1}}{12 (ab)^{-2}}$$. 2. Simplify the numerator: Multiply the coefficients and the variables separat

1. State the problem: Solve the equation $2x + 10 = 23$ for $x$. 2. Subtract 10 from both sides to isolate the term with $x$:

1. **Problem 19:** Given the data set 2, 4, 5, 8, 11, determine the mean, median, mode, range, variance, and standard deviation. 2. **Mean:** The mean is the average of the data po

1. The problem states that $y$ is inversely proportional to $x$, which means we can write the relationship as: $$y \propto \frac{1}{x}$$

1. The problem is to solve the equation $2x + 10 = 25$ for $x$. 2. Start by isolating the term with $x$ on one side. Subtract 10 from both sides:

1. The problem states that $d = \frac{2.8}{h}$ and asks for the value of $d$ when $h = 0.07$. 2. Substitute $h = 0.07$ into the formula:

1. Stating the problem: Simplify the expression $$\frac{-3a^2 \times 4 \times b^{-1}}{12 (ab)^{-2}}$$. 2. Simplify the numerator: Multiply the constants and variables.

1. **State the problem:** We have two types of printers: old and new.

1. **State the problem:** We want to solve the equation $$4(x-3)-3(x-5)=3(x+1)$$ and understand why the solution is $$x=0$$. 2. **Expand each term:**

1. **Problem statement:** A group of 12 people have enough water to last 9 days. (a) If 6 more people join, how long will the water last?

1. Problem 16a: Find values of $m$ such that the quadratic equation $$x^2 + (3 - m)x + 2m - 1 = 0$$ has real roots. 2. For real roots, the discriminant must be non-negative:

1. The problem is to analyze and correct the graph of the function $f(x) = 8 \cdot \left(\frac{5}{4}\right)^x$. 2. This is an exponential function with base $\frac{5}{4} > 1$, so t

1. **State the problem:** Solve the equation $$\frac{1}{x} - 1 = \frac{3e^{3x}}{x+3} - \frac{e^{3x}}{(x+3)^2}$$ for the values $x = -2.75$, $x = -2$, and $x = -1$. 2. **Substitute

1. **State the problem:** We have a quadratic function $f(x) = ax^2 + bx + c$ with vertex at $(0.5, -12.5)$ and x-intercepts at $(2, 0)$ and $(p, 0)$. We need to find $p$, the coef

1. **State the problem:** We have a cylindrical tin can with height function $h(t) = 3t^2 + 0.5$ for $4 \leq t \leq 14$. We want to find the height when $t$ is given. 2. **Calculat

1. The problem is to solve the equation $7 \frac{b+5}{10} = \frac{5}{7}$ for $b$. 2. Start by writing the equation clearly:

1. The problem is to solve the equation $\frac{7}{n} = \frac{8}{7}$ for $n$. 2. To solve for $n$, cross-multiply the fractions:

1. Stating the problem: Solve the equation $\frac{7}{b} + 5 = \frac{10}{5}$ for $b$. 2. Simplify the right side: $\frac{10}{5} = 2$.

1. Problem: Calculate the percentage error in measuring a string of length 400 cm when rounded to the nearest metre and nearest 100 m. Step 1: Convert 400 cm to metres: $$400\text{