Solve the first part of the inequality: 4 - 3x < x + 12.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Solve the first part of the inequality: $4 - 3x < x + 12$.
$$4 - 3x < x + 12$$
Subtract $x$ from both sides:
$$4 - 4x < 12$$
Subtract $4$ from both sides:
$$-4x < 8$$
Divide by $-4$. Remember to reverse the inequality sign when dividing by a negative number:
$$x > \frac{8}{-4}$$
$$x > -2$$
Step 2: Solve the second part of the inequality: $x + 12 \le \frac{-3x + 29}{2}$.
$$x + 12 \le \frac{-3x + 29}{2}$$
Multiply both sides by $2$:
$$2(x + 12) \le -3x + 29$$
$$2x + 24 \le -3x + 29$$
Add $3x$ to both sides:
$$5x + 24 \le 29$$
Subtract $24$ from both sides:
$$5x \le 5$$
Divide by $5$:
$$x \le 1$$
Step 3: Combine the solutions.
We have $x > -2$ and $x \le 1$. This can be written as:
$$-2 < x \le 1$$
Step 4: Illustrate the solution on a number line.
The solution includes all numbers greater than $-2$ and less than or equal to $1$. On a number line, this is represented by an open circle at $-2$ (since $-2$ is not included), a closed circle at $1$ (since $1$ is included), and a line segment connecting these two points.
`
<-----|-----|-----|-----|-----|-----|-----|----->
-3 -2 -1 0 1 2 3
o------------------•
`
The solution is $\boxed{-2 < x \le 1}$.
Step 1: Evaluate each term in the expression.
$$\left(\frac{1}{27}\right)^{\frac{1}{3}} = \left(\frac{1}{3^3}\right)^{\frac{1}{3}} = (3^{-3})^{\frac{1}{3}} = 3^{-3 \times \frac{1}{3}} = 3^{-1} = \frac{1}{3}$$
$$\sqrt{256 \times 9} = \sqrt{256} \times \sqrt{9} = 16 \times 3 = 48$$
$$729^{-1} = \frac{1}{729}$$
$$72^2 = (8 \times 9)^2 = 8^2 \times 9^2 = 64 \times 81$$
Step 2: Substitute the evaluated terms back into the expression.
$$\frac{\frac{1}{3} \times 48}{\frac{1}{729} \times 72^2}$$
Step 3: Simplify the numerator.
$$\frac{1}{3} \times 48 = 16$$
Step 4: Simplify the denominator.
$$\frac{1}{729} \times 72^2 = \frac{1}{9^3} \times (8 \times 9)^2 = \frac{1}{9^3} \times 8^2 \times 9^2 = \frac{8^2}{9} = \frac{64}{9}$$
Step 5: Divide the simplified numerator by the simplified denominator.
$$\frac{16}{\frac{64}{9}} = 16 \times \frac{9}{64}$$
$$ = \frac{16 \times 9}{64}$$
$$ = \frac{1 \times 9}{4}$$
$$ = \frac{9}{4}$$
The evaluated expression is $\boxed{\frac{9}{4}}$.
Step 1: Calculate the amount in Kenya Shillings (Ksh) the tourist received from changing South African Rands (SAR).
On 12/1/2024, the tourist sells SAR to the bank, so the bank buys SAR.
Buying rate for 1 SAR = 7.08 Ksh.
Amount in Ksh = $15\,000 \text{ SAR} \times 7.08 \text{ Ksh/SAR} = 106\,200 \text{ Ksh}$.
Step 2: Calculate the remaining Kenya Shillings after spending.
Amount spent = 53 075 Ksh.
Remaining Ksh = $106\,200 \text{ Ksh} - 53\,075 \text{ Ksh} = 53\,125 \text{ Ksh}$.
Step 3: Calculate the amount in Tanzanian Shillings (Tsh) the tourist received from changing the remaining Kenya Shillings.
On 12/2/2024, the tourist buys Tsh from the bank, so the bank sells Tsh.
Selling rate for 100 Tanzanian Shillings = 21.25 Ksh.
This means 1 Tsh = $\frac{21.25}{100} \text{ Ksh} = 0.2125 \text{ Ksh}$.
Amount in Tsh = $\frac{\text{Remaining Ksh}}{\text{Ksh per Tsh}}$
Amount in Tsh = $\frac{53\,125 \text{ Ksh}}{0.2125 \text{ Ksh/Tsh}} = 250\,000 \text{ Tsh}$.
The amount received in Tanzanian shillings is $\boxed{250\,000 \text{ Tsh}}$.
That's 2 down. 3 left today — send the next one.
