Question

1. geometry write and solve an equation to find the value of x so that the figure has the same area.
2. twice the greater integer of two consecutive odd integers is 13 less than three times the lesser integer.

a. write an equation to find the two consecutive odd integers.
b. what are the integers in ascending order?

Explanation:

14.

Step1: Write area formulas

The area formula of a triangle is \(A_{triangle}=\frac{1}{2}\times base\times height\). Here, base \(b = 8\) and height \(h=x + 4\), so \(A_{triangle}=\frac{1}{2}\times8\times(x + 4)=4(x + 4)\).
The area formula of a rectangle is \(A_{rectangle}=length\times width\). Here, length \(l=x + 6\) and width \(w = 3\), so \(A_{rectangle}=3(x + 6)\).

Step2: Set up the equation

Since the areas are equal, we set \(4(x + 4)=3(x + 6)\).
Expand both sides: \(4x+16 = 3x + 18\).

Step3: Solve the equation

Subtract \(3x\) from both sides: \(4x-3x+16=3x - 3x+18\), which gives \(x+16 = 18\).
Subtract \(16\) from both sides: \(x=18 - 16\).

Let the lesser odd integer be \(n\). Then the greater odd integer is \(n + 2\).
Twice the greater integer is \(2(n + 2)\), and three times the lesser integer minus \(13\) is \(3n-13\).
Since twice the greater integer of two - consecutive odd integers is \(13\) less than three times the lesser integer, the equation is \(2(n + 2)=3n-13\).

Step1: Expand the equation

Expand \(2(n + 2)=3n-13\) to get \(2n+4 = 3n-13\).

Step2: Solve for \(n\)

Subtract \(2n\) from both sides: \(2n-2n + 4=3n-2n-13\), which gives \(4=n - 13\).
Add \(13\) to both sides: \(n=4 + 13=17\).
The greater odd integer is \(n + 2=17+2 = 19\).

Answer:

\(x = 2\)

15.

a.