Solve: 1/3(x−2)

Question

Solve: 1/3(x−2) <= 1/5(2x+3)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

1. Solve: $\frac{1}{3}(x-2) \le \frac{1}{5}(2x+3)$

Multiply both sides by the least common multiple of the denominators (3 and 5), which is 15.
$15 \cdot \frac{1}{3}(x-2) \le 15 \cdot \frac{1}{5}(2x+3)$ $5(x-2) \le 3(2x+3)$
Distribute the numbers on both sides.
$5x - 10 \le 6x + 9$
Gather $x$ terms on one side and constant terms on the other.
$-10 - 9 \le 6x - 5x$ $-19 \le x$
This can also be written as $x \ge -19$ .

The correct option is C.
$\boxed{\text{C. } x \ge -19}$
2. The ages in years of six teachers in a school are: $3x, 42, (2x+6), 28, 49, 31$ . If the mean age is 36 years, find the value of $x$ .
Write the formula for the mean.
$\text{Mean} = \frac{\text{Sum of ages}}{\text{Number of teachers}}$
Substitute the given ages and mean into the formula.
$36 = \frac{3x + 42 + (2x+6) + 28 + 49 + 31}{6}$
Simplify the numerator by combining like terms.
$36 = \frac{(3x + 2x) + (42 + 6 + 28 + 49 + 31)}{6}$ $36 = \frac{5x + 156}{6}$
Solve for $x$ .
Multiply both sides by 6.
$36 \times 6 = 5x + 156$ $216 = 5x + 156$
Subtract 156 from both sides.
$216 - 156 = 5x$ $60 = 5x$
Divide by 5.
$x = \frac{60}{5}$ $x = 12$
The correct option is C.
$\boxed{\text{C. } 12}$
3. Given that $2x - y = 11$ and $4x + y = 1$ , find the value of $(x+y)$ .
Use the elimination method to solve the system of equations.
Add the two equations together.
$(2x - y) + (4x + y) = 11 + 1$ $6x = 12$
Solve for $x$ .
$x = \frac{12}{6}$ $x = 2$
Substitute the value of $x$ into the second equation ( $4x + y = 1$ ) to find $y$ .
$4(2) + y = 1$ $8 + y = 1$ $y = 1 - 8$ $y = -7$
Calculate the value of $(x+y)$ .
$x+y = 2 + (-7)$ $x+y = -5$
The correct option is A.
$\boxed{\text{A. } -5}$
4. The mean of $4, 5, 7, 8$ and 11 is 7. Calculate the standard deviation.
List the data points ( $x_i$ ) and the mean ( $\bar{x}$ ).
Data points: $4, 5, 7, 8, 11$
Mean ( $\bar{x}$ ): 7
Number of data points ( $n$ ): 5
Calculate the deviation of each data point from the mean ( $x_i - \bar{x}$ ) and square it $(x_i - \bar{x})^2$ .
- $4 - 7 = -3 \implies (-3)^2 = 9$
- $5 - 7 = -2 \implies (-2)^2 = 4$
- $7 - 7 = 0 \implies (0)^2 = 0$
- $8 - 7 = 1 \implies (1)^2 = 1$
- $11 - 7 = 4 \implies (4)^2 = 16$
Sum the squared deviations.
$\sum (x_i - \bar{x})^2 = 9 + 4 + 0 + 1 + 16 = 30$
Calculate the variance ( $\sigma^2$ ) using the formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ .
$\sigma^2 = \frac{30}{5} = 6$
Calculate the standard deviation ( $\sigma$ ) by taking the square root of the variance.
$\sigma = \sqrt{6}$
The correct option is B.
$\boxed{\text{B. } \sqrt{6}}$
5. Given that $\vec{OT} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$ , $\vec{AB} = \begin{pmatrix} -1 \\ -2 \end{pmatrix}$ , where $O$ is the origin, find $\vec{OB}$ .
Understand the vector relationship.
The vector $\vec{AB}$ can be expressed as $\vec{OB} - \vec{OA}$ .
$\vec{AB} = \vec{OB} - \vec{OA}$
Since $O$ is the origin, $\vec{OA}$ is the position vector of point A. The vector $\vec{OT}$ is a distractor in this problem, as it refers to point T, which is not involved in finding $\vec{OB}$ from $\vec{AB}$ .
To find $\vec{OB}$ , we need $\vec{OA}$ .
Given the multiple-choice options, the most plausible interpretation is that point A coincides with the origin $O$ . If $A$ is the origin, then its position vector $\vec{OA}$ is the zero vector.
$\vec{OA} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$
Substitute $\vec{OA}$ into the vector relationship.
$\vec{AB} = \vec{OB} - \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ $\vec{AB} = \vec{OB}$
Use the given value for $\vec{AB}$ to find $\vec{OB}$ .
$\vec{OB} = \begin{pmatrix} -1 \\ -2 \end{pmatrix}$
The correct option is A.
$\boxed{\text{A. } \begin{pmatrix} -1 \\ -2 \end{pmatrix}}$

Solve: 1/3(x-2) <= 1/5(2x+3)

Answer

Step-by-step solution

Multiply both sides by the least common multiple of the denominators (3 and 5), which is 15.

Distribute the numbers on both sides.

Gather $x$ terms on one side and constant terms on the other.

Write the formula for the mean.

Substitute the given ages and mean into the formula.

Simplify the numerator by combining like terms.

Solve for $x$ .

Use the elimination method to solve the system of equations.

Solve for $x$ .

Substitute the value of $x$ into the second equation ( $4x + y = 1$ ) to find $y$ .

Calculate the value of $(x+y)$ .

List the data points ( $x_i$ ) and the mean ( $\bar{x}$ ).

Calculate the deviation of each data point from the mean ( $x_i - \bar{x}$ ) and square it $(x_i - \bar{x})^2$ .

Sum the squared deviations.

Calculate the variance ( $\sigma^2$ ) using the formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ .

Calculate the standard deviation ( $\sigma$ ) by taking the square root of the variance.

Understand the vector relationship.

To find $\vec{OB}$ , we need $\vec{OA}$ .

Substitute $\vec{OA}$ into the vector relationship.

Use the given value for $\vec{AB}$ to find $\vec{OB}$ .

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More Mathematics Questions

Solve: 1/3(x-2) <= 1/5(2x+3)

Answer

Step-by-step solution

Multiply both sides by the least common multiple of the denominators (3 and 5), which is 15.

Distribute the numbers on both sides.

Gather xxx terms on one side and constant terms on the other.

Write the formula for the mean.

Substitute the given ages and mean into the formula.

Simplify the numerator by combining like terms.

Solve for xxx.

Use the elimination method to solve the system of equations.

Solve for xxx.

Substitute the value of xxx into the second equation (4x+y=14x + y = 14x+y=1) to find yyy.

Calculate the value of (x+y)(x+y)(x+y).

List the data points (xix_ixi​) and the mean (xˉ\bar{x}xˉ).

Calculate the deviation of each data point from the mean (xi−xˉx_i - \bar{x}xi​−xˉ) and square it (xi−xˉ)2(x_i - \bar{x})^2(xi​−xˉ)2.

Sum the squared deviations.

Calculate the variance (σ2\sigma^2σ2) using the formula σ2=∑(xi−xˉ)2n\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}σ2=n∑(xi​−xˉ)2​.

Calculate the standard deviation (σ\sigmaσ) by taking the square root of the variance.

Understand the vector relationship.

To find OB⃗\vec{OB}OB, we need OA⃗\vec{OA}OA.

Substitute OA⃗\vec{OA}OA into the vector relationship.

Use the given value for AB⃗\vec{AB}AB to find OB⃗\vec{OB}OB.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Gather $x$ terms on one side and constant terms on the other.

Solve for $x$ .

Solve for $x$ .

Substitute the value of $x$ into the second equation ( $4x + y = 1$ ) to find $y$ .

Calculate the value of $(x+y)$ .

List the data points ( $x_i$ ) and the mean ( $\bar{x}$ ).

Calculate the deviation of each data point from the mean ( $x_i - \bar{x}$ ) and square it $(x_i - \bar{x})^2$ .

Calculate the variance ( $\sigma^2$ ) using the formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ .

Calculate the standard deviation ( $\sigma$ ) by taking the square root of the variance.

To find $\vec{OB}$ , we need $\vec{OA}$ .

Substitute $\vec{OA}$ into the vector relationship.

Use the given value for $\vec{AB}$ to find $\vec{OB}$ .