A meteorologist announces that yesterday’s high temperature was within 3° Celsius of 20° Celsius all across the country. Write an inequality that involves an absolute value to represent yesterday’s high temperature. Solve the inequality and explain the meaning of the solution.
Let x be the high temperature yesterday.
The given information states that yesterday's high temperature was within 3° Celsius of 20° Celsius. This can be represented using the absolute value of the difference between x and 20, which should be less than or equal to 3. Therefore, the inequality that represents yesterday's high temperature is:
|x - 20| ≤ 3
To solve this inequality, we need to find the values of x that make the absolute value expression less than or equal to 3. We can do this by considering two cases:
1. Case: (x - 20) ≤ 0
In this case, the absolute value is equal to x - 20, and the inequality becomes:
x - 20 ≤ 3
Adding 20 to both sides, we get:
x ≤ 23
2. Case: (x - 20) > 0
In this case, the absolute value is equal to -(x - 20), and the inequality becomes:
-(x - 20) ≤ 3
Expanding the inequality by distributing the negative sign, we get:
- x + 20 ≤ 3
Subtracting 20 from both sides, we get:
- x ≤ -17
Multiplying both sides by -1 and changing the direction of the inequality, we get:
x ≥ 17
Combining the solutions from both cases, the valid values for x are:
17 ≤ x ≤ 23
Therefore, the meaning of the solution is that yesterday's high temperature across the country was between 17° Celsius and 23° Celsius, inclusive.
To represent yesterday's high temperature using an absolute value, we can write the inequality as follows:
|y - 20| ≤ 3
The absolute value represents the difference between the actual temperature (y) and 20 degrees Celsius. The inequality states that the absolute value of this difference must be less than or equal to 3.
To solve the inequality, we can consider two cases:
Case 1: (y - 20) ≥ 0
In this case, the absolute value equation simplifies to y - 20 ≤ 3. Solving for y, we add 20 to both sides:
y ≤ 23
Case 2: (y - 20) < 0
In this case, the absolute value equation simplifies to -(y - 20) ≤ 3. This can be rewritten as -y + 20 ≤ 3. Solving for y, we subtract 20 from both sides and multiply by -1 while reversing the inequality sign:
y ≥ 17
Combining the solutions from both cases, we find that the temperature range that satisfies the given condition is 17 ≤ y ≤ 23. This means that yesterday's high temperature could have been any value between 17 and 23 degrees Celsius, inclusive.
To represent yesterday's high temperature using an absolute value, we can use the following inequality:
|T - 20| ≤ 3
Here, T represents the actual high temperature. The equation states that the absolute value of the difference between T and 20 should be less than or equal to 3.
To solve this inequality, we can consider two cases:
Case 1: T - 20 ≥ 0 (when T is greater than or equal to 20)
If T is greater than or equal to 20, the inequality becomes:
T - 20 ≤ 3
Adding 20 to both sides, we have:
T ≤ 23
So, if the temperature is greater than or equal to 20, it must be less than or equal to 23.
Case 2: T - 20 < 0 (when T is less than 20)
If T is less than 20, the inequality becomes:
-(T - 20) ≤ 3
Which simplifies to:
- T + 20 ≤ 3
Subtracting 20 from both sides, we get:
- T ≤ -17
Multiplying both sides by -1 (we must change the inequality direction when multiplying/dividing by a negative number), we have:
T ≥ 17
So, if the temperature is less than 20, it must be greater than or equal to 17.
Combining both cases, the overall solution is:
17 ≤ T ≤ 23
The meaning of this solution is that the high temperature yesterday was within the range of 17 to 23 degrees Celsius.