find all numbers such that a third of a number increased by half that number is at least 3 less than that same number.
not sure what to do
2020 was a hell of a year
x/3 + x/2 >= x-3
2x + 3x >= 6x-18
-x >= -18
x <= 18
Well, numbers can be quite mischievous sometimes! Let's try to solve this riddle together.
Let's call the number we're trying to find "x". According to the problem, a third of x increased by half of x should be at least 3 less than x. Mathematically, this can be written as:
(1/3)x + (1/2)x ≥ x - 3
Simplifying this equation, we get:
(5/6)x ≥ x - 3
Let's subtract x from both sides:
(5/6)x - x ≥ -3
Multiplying both sides by 6 to get rid of the fraction, we have:
5x - 6x ≥ -18
-1x ≥ -18
Dividing both sides by -1 (and remember, dividing by a negative number flips the inequality sign!), we get:
x ≤ 18
So, all numbers less than or equal to 18 satisfy the given condition. Let's keep our eyes open for those prankster numbers!
To solve this problem, let's break it down into smaller parts:
Let's assume the number is represented by "x".
According to the given condition, "a third of a number increased by half that number is at least 3 less than that same number," we can write the following equation:
(1/3)x + (1/2)x ≤ x - 3
To simplify the equation, we need to find a common denominator for (1/3) and (1/2), which is 6:
(2/6)x + (3/6)x ≤ x - 3
Combining like terms:
(5/6)x ≤ x - 3
To eliminate fractions, we can multiply both sides of the equation by 6:
6 * (5/6)x ≤ 6 * (x - 3)
5x ≤ 6x - 18
Next, we can isolate the variable on one side:
5x - 6x ≤ -18
-x ≤ -18
Now, let's multiply both sides by -1 (which also reverses the direction of the inequality):
x ≥ 18
Therefore, any number greater than or equal to 18 satisfies the given condition.
To find the numbers that satisfy the given condition, we need to set up an equation based on the problem statement. Let's break down the problem step by step and determine the equation to solve it.
Step 1: "A third of a number increased by half that number."
Let's represent the number as "x". "A third of a number" can be expressed as (1/3)x, and "half that number" is (1/2)x. So, according to the statement, we can write the equation as:
(1/3)x + (1/2)x
Step 2: "...is at least 3 less than that same number."
To check if it is "at least 3 less than that same number," we subtract 3 from x. So the equation becomes:
(1/3)x + (1/2)x ≤ x - 3
Step 3: Solve the equation
To solve the equation, we multiply through by a common denominator to eliminate fractions. The common denominator between 3 and 2 is 6, so multiply everything by 6 to clear the fractions:
6 * (1/3)x + 6 * (1/2)x ≤ 6 * (x - 3)
2x + 3x ≤ 6x - 18
5x ≤ 6x - 18
5x - 6x ≤ -18
-x ≤ -18
x ≥ 18
Therefore, any number equal to or greater than 18 satisfies the given condition.
In conclusion, all numbers greater than or equal to 18 are the solutions.