Sure! Let's break down and solve each of the given equations and the inequality step by step.
### Step Equations
#### 1. Solve for [tex]\( y \)[/tex]:
Given:
[tex]\[ \frac{y}{6} - 7 = 4 \][/tex]
1. First, add 7 to both sides to isolate the fraction:
[tex]\[ \frac{y}{6} = 4 + 7 \][/tex]
[tex]\[ \frac{y}{6} = 11 \][/tex]
2. Next, multiply both sides by 6 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 11 \times 6 \][/tex]
[tex]\[ y = 66 \][/tex]
So, [tex]\( y = 66 \)[/tex].
#### 2. Solve for [tex]\( g \)[/tex]:
Given:
[tex]\[ \frac{g}{3} + 11 = 25 \][/tex]
1. First, subtract 11 from both sides to isolate the fraction:
[tex]\[ \frac{g}{3} = 25 - 11 \][/tex]
[tex]\[ \frac{g}{3} = 14 \][/tex]
2. Next, multiply both sides by 3 to solve for [tex]\( g \)[/tex]:
[tex]\[ g = 14 \times 3 \][/tex]
[tex]\[ g = 42 \][/tex]
So, [tex]\( g = 42 \)[/tex].
#### 3. Solve for [tex]\( p \)[/tex]:
Given:
[tex]\[ -4p + 19 = 11 \][/tex]
1. First, subtract 19 from both sides to isolate the term with [tex]\( p \)[/tex]:
[tex]\[ -4p = 11 - 19 \][/tex]
[tex]\[ -4p = -8 \][/tex]
2. Next, divide both sides by -4 to solve for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{-8}{-4} \][/tex]
[tex]\[ p = 2 \][/tex]
So, [tex]\( p = 2 \)[/tex].
### Step Inequalities
Given the following inequality statement:
"Duncan scored at least 27 points more than half of what Duncan scored. Write an inequality that could be solved to find Duncan's score."
1. Let [tex]\( D \)[/tex] represent Duncan's score.
2. The inequality described is:
[tex]\[ D \geq \frac{D}{2} + 27 \][/tex]
Let's solve for [tex]\( D \)[/tex]:
1. Start by subtracting [tex]\(\frac{D}{2}\)[/tex] from both sides to get all the terms involving [tex]\( D \)[/tex] on one side:
[tex]\[ D - \frac{D}{2} \geq 27 \][/tex]
2. This simplifies to:
[tex]\[ \frac{D}{2} \geq 27 \][/tex]
3. Multiply both sides by 2 to isolate [tex]\( D \)[/tex]:
[tex]\[ D \geq 27 \times 2 \][/tex]
[tex]\[ D \geq 54 \][/tex]
So, Duncan must have scored at least 54 points.
This provides [tex]\( D \geq 54 \)[/tex] as the solution for the inequality. However, based on the given results, Duncan might have scored exactly 54, so [tex]\( D = -48 \)[/tex] does not relate to our consistent given question. We can consult these steps for clarity.
