Answer:
x > 2
Explanation:
Step 1: Find area of the rectangle.
Length of rectangle = 5x - 6
Width of rectangle = x - 1
Area of rectangle = length*width
Area of rectangle = (5x - 6)(x - 1)
= 5x(x - 1) -6(x - 1)
= 5x² - 5x - 6x + 6
Area of rectangle = 5x² - 11x + 6
Step 2: Find are of the triangle
Area of triangle = ½*base*height
Base of triangle = x
Height of triangle = 2x
Area = ½*x*2x
Area = x*x
Area of triangle = x²
Step 3: Write the inequality statement for "the area of the rectangle is greater than the area of the triangle", and solve for possible values of x.
Area of rectangle = 5x² - 11x + 6
Area of triangle = x²
Therefore:
5x² - 11x + 6 > x²
Subtract x² from both sides
5x² - 11x + 6 - x² > x² - x²
5x² - x² - 11x + 6 > 0
4x² - 11x + 6 > 0
Factorise (4x² - 11x + 6)
(The 2 possible factors when multiplied together will give us "24", and when added together will give us "-11" are -8 and -3).
Therefore, we factorise as follows:
4x² - 8x - 3x + 6 > 0
(4x² - 8x) - (3x + 6) > 0
4x(x - 2) -3(x - 2) > 0
(4x - 3)(x - 2) > 0
Find the solution of each factor of the inequality
(4x - 3) > 0
Or
(x - 2) > 0
Thus, solving for each for each, we have:
4x - 3 > 0
Add 3 to both sides
4x - 3 + 3 > 0 + 3
4x > 3
Divide both sides by 4
4x/4 > 3/4
x > ¾
Or
x - 2 > 0
Add 2 to both sides
x - 2 + 2 > 0 + 2
x > 2
x > ¾ or x > 2
The possible values of x that fits the inequality given is x > 2, rather than x > ¾
Rationale:
We are told that the area of rectangle > area of triangle. Thus, the inequality given is:
5x² - 11x + 6 > x²
If x > ¾, that is the first possible value of x is 1, plug in the value of x into the inequality let's find out if it is true for the statement.
Thus:
5x² - 11x + 6 > x²
5(1)² - 11(1) + 6 > 1²
5 - 11 + 6 > 1
0 > 1 (this is very untrue)
Let's try out x > 2, meaning the first possible value of x = 3
Thus,
5x² - 11x + 6 > x²
5(3)² - 11(3) + 6 > 3²
5(9) - 33 + 6 > 9
45 - 33 + 6 > 9
18 > 9 (very true)
Therefore, the possible value of x would be values of x that are greater than 2.
Our answer is x > 2.