🧮 algebra

1. The problem involves graphs of $f(x) = \frac{1}{2}(x+p)^2 + q$ and $g(x) = \frac{a}{x+r} + t$, with axis $h(x) = -x + 3$ symmetrical to $g$, and points labeled as C, E, B among

Solve the following exponential equations and inequalities step-by-step. 1. Solve $4^{3x+1} = 8^{x-1}$:

1. **State the problem:** Solve the equation $24x + 18 - 4x = 21x - 12$ for $x$. 2. **Simplify the left side:** Combine like terms $24x - 4x$ to get $20x$, so the equation becomes:

1. The problem is to evaluate the expression $2^3 + 2^1$. 2. Calculate $2^3$: since $2^3 = 2 \times 2 \times 2 = 8$.

1. The problem is to simplify the expression $$\left(\frac{2}{3}\right)^{-3}$$ and verify that it equals $$\left(\frac{3}{2}\right)^3$$ and $$\frac{27}{8}$$. 2. Start by applying t

1. **Stating the problem:** A school bag is on sale at 15% off, and the sale price is 841.50. We need to find the original price before the discount. 2. **Understanding the discoun

1. The problem involves evaluating two expressions: $(-3)^4$ and $(-2)^3$. 2. First, calculate $(-3)^4$. This means multiplying $-3$ by itself 4 times:

1. The problem is to simplify expressions or equations by eliminating fractions. 2. To remove a fraction from an equation, multiply both sides of the equation by the denominator to

1. First, state the problem: evaluate $8^3 \times 4^{-2}$. 2. Calculate $8^3$: since $8 = 2^3$, then $8^3 = (2^3)^3 = 2^{9} = 512$.

1. The problem is to draw the curve of the function $y=-(x-5)-2$. 2. Start by simplifying the expression inside the function.

1. Let's first clarify the problem statement. You have an expression where the square root applies only to the squared part of it, not the entire expression. 2. For example, if the

1. Let's analyze the sequence given starting from the simple arithmetic statement $1 + 1 = 2$. 2. Next steps attempt to transform or equate $2$ to other expressions such as $4 - 2$

1. State the problem: Simplify the expression $$H = \sqrt{3} - 2\sqrt{2} + \sqrt{\frac{3}{2}} \sqrt{9}$$. 2. Simplify the term $$\sqrt{\frac{3}{2}} \sqrt{9}$$ using the property $$

1. Let $f(x) = x^2 - 1$ and $g(x) = 2x + 3$. Find; (a) $(f \circ g)(x)$ and $(g \circ f)(x)$.

1. Problem a: Given $a^x = b^y = c^z$ and $a, b, c$ are in geometric progression (G.P.), prove that $x, y, z$ are in harmonic progression (H.P.). Step 1: Since $a, b, c$ are in G.P

1. **Problem 1: Calculate the discount and sale price for a knapsack originally priced at 2500 with a 20% discount.** Step 1: Calculate the discount amount.

1. State the problem: Prove that $-(-a) = a$ for any number $a$. 2. Understand the sign notation: The negative sign $-$ before a number means the additive inverse of that number.

1. The problem asks for the percent of change when the alcohol volume changes from 78 ml to 96.72 ml. 2. Percent change formula is $$\text{Percent Change} = \frac{\text{New Value}

1. Problem: Calculate the 5% discount on a cellular phone worth 12,690. Step 1: Calculate discount = $5\% \times 12,690 = 0.05 \times 12,690$

1. The problem is to find the percent of change in enrolment from the previous year to this year. 2. Let the previous year's enrolment be $P$ and this year's enrolment be $T$.

1. **Problem 1:** Write the equation for the joint variation "x varies jointly as y and z". Since x varies jointly as y and z, the formula is: