twice the difference of a number and 9 is at most -22
x = your number
2 ( x - 9 ) ≤ - 22
2 x - 18 ≤ - 22
Add 18 to both sides
2 x ≤ - 4
Divide both sides by 2
x ≤ - 2
2(x-9) ≤ -22
2x - 18 ≤ -22
2x ≤ -4
x ≤ -2
but the difference between a number and 9 could be
either x-9 or 9-x
Can we have 2(9-x) ≤ -22 ?
18 - 2x ≤ -22
-2x ≤ -40
x ≥ 20
if we took the difference as x -9 then x ≤ -2
if we took the difference as 9-x then x ≥ 20
Let's assign a variable to represent the number. Let's call it "x".
The statement "twice the difference of a number and 9 is at most -22" can be written as:
2(x - 9) ≤ -22
Now, let's solve this inequality step-by-step:
1. Distribute the 2:
2x - 18 ≤ -22
2. Add 18 to both sides to isolate the variable term:
2x ≤ -22 + 18
Simplifying:
2x ≤ -4
3. Divide both sides of the inequality by 2 to solve for x:
x ≤ -4/2
Simplifying:
x ≤ -2
Therefore, the solution to the inequality "twice the difference of a number and 9 is at most -22" is x ≤ -2.
To solve this problem, let's break it down into steps:
Step 1: Let's assume the number we are looking for is "x".
Step 2: The difference between a number and 9 is given by |x - 9| (absolute value).
Step 3: Twice the difference is 2 * |x - 9|.
Step 4: According to the problem, the expression "2 * |x - 9|" is at most -22. We can write this as an inequality: 2 * |x - 9| ≤ -22.
Step 5: To solve this inequality, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
Case 1: (x - 9) ≥ 0 (x is greater than or equal to 9)
In this case, the absolute value can be removed: 2(x - 9) ≤ -22.
Simplifying the inequality gives: 2x - 18 ≤ -22.
Adding 18 to both sides: 2x ≤ -4.
Dividing by 2: x ≤ -2.
Case 2: (x - 9) < 0 (x is less than 9)
Here, we change the sign of the inequality when removing the absolute value: 2(-x + 9) ≤ -22.
Simplifying the inequality gives: -2x + 18 ≤ -22.
Subtracting 18 from both sides: -2x ≤ -40.
Dividing by -2 (remember to flip the inequality sign): x ≥ 20.
Step 6: Now, we need to consider the values that satisfy both cases. From case 1, we have x ≤ -2, and from case 2, we have x ≥ 20. The only value that satisfies both cases is x = 20.
Therefore, the number we are looking for is 20.