Solve the inequality. Write the solution in interval notation.
(x+14)/(x-8)≥ 1
if x-8> 0 , then
x+14 ≥ x-8
14 ≥ -8 , so true for all x's
if x-8 < 0
x+14 ≤ x-8
14 ≤ -8 , no way!
so true for x > 8
check:
take a value > 8
e.g. x = 10
statement becomes
24/2 ≥ 1 , true
take a value < 8
e.g
x = 1
statement becomes
15/-7≥1 , false
how about x = -20
statement becomes
-6/-28 ≥ 1 or 3/14 ≥ 1 false
solution x > 8
To solve the inequality: (x + 14)/(x - 8) ≥ 1
First, let's find the values of x that make the left side of the inequality equal to 1.
(x + 14)/(x - 8) = 1
To remove the fraction, we can multiply both sides of the equation by (x - 8):
(x + 14) = (x - 8)
Expanding the equation gives:
x + 14 = x - 8
Now, we can solve for x:
x - x = -8 - 14
0 = -22
This equation has no solutions. Therefore, there are no values of x that make the left side of the equation equal to 1.
Now, let's consider the inequality sign and find the intervals where the inequality is satisfied.
Since there are no solutions to the equation, the fraction (x + 14)/(x - 8) will be greater than 1 for all valid values of x.
Therefore, the solution to the inequality is x ∈ (-∞, ∞).
To solve the inequality (x+14)/(x-8)≥ 1, we need to find the values of x that make the inequality true. Here's how we can do it step by step:
Step 1: Set the equation equal to 1
(x+14)/(x-8) = 1
Step 2: Multiply both sides of the equation by (x-8) to remove the denominator
(x+14) = (x-8)
Step 3: Expand the equation
x + 14 = x - 8
Step 4: Move all the terms involving x to one side of the equation
x - x = -8 - 14
Step 5: Simplify
0 = -22
Step 6: Check if the equation is true or false
Since 0 does not equal -22, this means that there is no value of x that satisfies the equation (x+14)/(x-8) = 1.
Therefore, there is no solution to the inequality (x+14)/(x-8) ≥ 1.
In interval notation, we can write this as an empty set:
∅