Question

1. 3(1 - x)+10x≤9(x - 2)+7
2. the students in charge of the class booth at a carnival want to make a profit of at least $65 by selling items. they spent $55 for the materials to make the items. they will sell each item for $3. find the possible values of x, the number of items the students need to sell to reach their goal.
3. a triangle has a base of 14 centimeters and a height of (3x - 4) centimeters. the area of the triangle is greater than 56 centimeters. find the possible values of x

Explanation:

Step1: Solve the inequality in question 12

First, expand the left - hand side of the inequality $3(1 - x)+10x\leq9(x - 2)+7$.
Expand $3(1 - x)$ to get $3-3x$ and $9(x - 2)$ to get $9x-18$.
The inequality becomes $3-3x + 10x\leq9x-18 + 7$.
Combine like terms: $3 + 7x\leq9x-11$.
Subtract $7x$ from both sides: $3\leq9x-7x-11$.
Simplify to $3\leq2x-11$.
Add 11 to both sides: $3 + 11\leq2x$, so $14\leq2x$.
Divide both sides by 2: $x\geq7$.

Step2: Solve the profit - related problem in question 13

The cost of making items is $55$ and they want a profit of at least $65$. They sell each item for $3$. Let the number of items be $x$.
The revenue from selling $x$ items is $3x$.
The profit formula is $Profit=Revenue - Cost$.
We want $3x-55\geq65$.
Add 55 to both sides: $3x\geq65 + 55$, so $3x\geq120$.
Divide both sides by 3: $x\geq40$.

Step3: Solve the triangle area problem in question 14

The area formula for a triangle is $A=\frac{1}{2}bh$, where $b = 14$ and $h=(3x - 4)$.
We know that $A>56$.
Substitute the values into the area formula: $\frac{1}{2}\times14\times(3x - 4)>56$.
First, simplify $\frac{1}{2}\times14 = 7$.
The inequality becomes $7(3x - 4)>56$.
Divide both sides by 7: $3x-4>8$.
Add 4 to both sides: $3x>8 + 4$, so $3x>12$.
Divide both sides by 3: $x > 4$.

Answer:

For question 12: $x\geq7$
For question 13: $x\geq40$
For question 14: $x>4$