For the inequality: 4𝑥^2 − 28𝑥 + 49 ≤ 0
a) Solve by either the graphing or the number-line method. (4 marks)
To solve the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0, we can first find the critical points by setting the expression equal to zero.
4𝑥^2 − 28𝑥 + 49 = 0
We can factor this quadratic expression:
(2𝑥 − 7)(2𝑥 − 7) = 0
From this, we can see that the critical point is 𝑥 = 7/2.
Next, we can plot this critical point on a number line and test a point in each of the three intervals created: 𝑥 < 7/2, 𝑥 = 7/2, and 𝑥 > 7/2.
Testing 𝑥 = 0:
4(0)^2 − 28(0) + 49 = 0 + 0 + 49 = 49 > 0
Testing 𝑥 = 7/2:
4(7/2)^2 − 28(7/2) + 49 = 49/2 - 98/2 + 49 = 49/2 - 98/2 + 98/2 = 49/2 > 0
Testing 𝑥 = 10:
4(10)^2 − 28(10) + 49 = 400 - 280 + 49 = 400 - 280 + 49 = 169 > 0
From these tests, we can see that the quadratic expression is positive for all values of 𝑥. Therefore, there are no solutions to the inequality.
So, the solution to the inequality 4𝑥^2 − 28𝑥 + 49 ≤ 0 is 𝑥 does not belong to any real numbers.