What is the greatest integer that satisfies $6 - x > -9$ ?
Got this from AoPs btw
adding x+9 ... $15 > x
looks like 15
6-x > -9
-x > -15
then
x < 15
So the greatest integer for the above is 14
Well, let's solve the inequality step by step:
$6 - x > -9$
First, let's subtract 6 from both sides:
$-x > -15$
Next, we need to multiply both sides by -1, but remember, when you multiply or divide an inequality by a negative number, you have to flip that lovely inequality sign! So we get:
$x < 15$
Now, we need to find the greatest integer that satisfies this inequality. Well, the largest integer less than 15 is 14. So the greatest integer that satisfies $6 - x > -9$ is 14. Why? Because 14 is less than 15, which makes the inequality true. Ta-da! 🎉
To find the greatest integer that satisfies the inequality $6 - x > -9$, we can solve the inequality by isolating $x$.
Let's start by subtracting 6 from both sides of the inequality:
$6 - x - 6 > -9 - 6$
This simplifies to:
$-x > -15$
Next, we multiply both sides of the inequality by -1. However, when we multiply or divide an inequality by a negative number, the direction of the inequality sign switches. So, we need to reverse the inequality:
$-x < 15$
Now, divide both sides of the inequality by -1, remembering to flip the inequality sign again:
$x > -15$
So, the greatest integer that satisfies the inequality $6 - x > -9$ is $x = -14$.