what is their common domain and range of y≤1,y>-2/3x-2 and y>-2/3x-2?
sorry - the range cannot include x.
also, why do you have "y > -2/3 x - 2" twice?
since the sloping line has range of all real numbers, the common range is just (-∞,1]
You're right, my mistake. The range cannot include x.
And sorry for the confusion. I had accidentally repeated the second inequality, which is the equation of a line. The common range is the intersection of the range of y ≤ 1 and the range of y > -2/3x - 2 line. Since y ≤ 1 constrains the range to be less than or equal to 1 and y > -2/3x - 2 can take any value greater than -2/3x - 2, the common range is (-∞, 1]. Thank you for pointing that out.
To find the common domain and range of the given three inequalities, let's analyze each one separately:
1. y ≤ 1: This inequality states that the y-values are less than or equal to 1. Therefore, the range of this inequality is (-∞, 1]. However, since there are no restrictions on the x-values, the domain is (-∞, ∞).
2. y > -2/3x - 2: This inequality represents a linear function with a slope of -2/3 and a y-intercept of -2. The range for this inequality is ( -∞, ∞) since there are no restrictions on the y-values. For the domain, we need to consider the x-values for which the line exists. Since it is a linear function, the domain is (-∞, ∞).
3. y > -2/3x - 2: This is the same inequality as the previous one. Hence, the domain and range for this inequality are the same as mentioned above.
To find the common domain, we need to identify the values of x for which all three inequalities exist. Since all three inequalities have a domain of (-∞, ∞), the common domain is also (-∞, ∞).
Similarly, to find the common range, we need to determine the values of y that satisfy all three inequalities. Since the first inequality has a maximum value of 1, and the other two inequalities have an unlimited range, the common range is (-∞, ∞).
In summary:
Common domain: (-∞, ∞)
Common range: (-∞, ∞)