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.

Answer 1

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.