In a certain town the temperature, x in degrees Celsius on a certain day is described by two statements:
If 3 times the temperature is increased by 2, the temperature is still less than 14°C.
Twice the temperature minus 7 is greater than -11°C.
Part A: Create a compound inequality to represent the temperature range. (3 points)
Part B: Can the temperature in this town be 5°C? Justify your answer by solving the inequalities in Part A. (3 points)
Part C: The average temperature in another town is 3°C, but the actual temperature is within 4°C of the average. Write and
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A compound inequality is one which contains not just one inequality sign, but two. For example, you may be familiar with the notation:
\[x < 2\]
meaning that we know that our unknown value 'x' is less than 2. However, let's say that we also know that for this same value of 'x':
\[x > 1\]
(i.e. the value of 'x' is greater than 1)
Combining these two pieces of information, we can use the following notation:
\[1 < x < 2\]
meaning that we know that the value of 'x' is between that of 1 and 2. This would be a compound inequality, and that's what we're looking to set up in Part A.
Part A: We're told in the question to let x be the temperature on a certain day in a certain town (in degree Celsius). Were also told:
1.)If 3 times the temperature is increased by 2, the temperature is still less than 14°C.
2.)Twice the temperature minus 7 is greater than -11°C.
We can represent these two pieces of information as two separate algebraic inequalities in terms of our unknown value x:
\[1.)3x + 2 < 14\]
\[2.) 2x - 7 > -11\]
So, to combine these and form our compound inequality, we'll try and get 'x' on its own in the two individually first. In doing, this, we can simply treat the '<' and '>' signs as if they were an '=' sign in this case, like if we were dealing with and equation.
1.) 3x + 2 < 14
Subtracting 2 from both sides, we get:
3x < 14 - 2, implying
3x < 12
Now, dividing each side by 3 to get x on its own, we're left with:
x < 4
That's one half of our compound inequality completed.
2.) 2x - 7 > -11
Getting x on its own as in 1.) above:
2x > -11 + 7, implying
2x > -4, implying
x > -2
So, we now know that the temperature in the town on a given day will be less that 4 degrees Celsius AND greater than -2 degrees Celsius.
Combining these pieces of information together, we're left with the following compound inequality for x (i.e. the range of possible temperatures in the town):
\[-2 < x < 4\]
Part B: The final compound inequality we've created in Part A. should automatically tell us whether a temperature of 5 degrees is possible, based on the information we're given.
Part C: The last part of the question appears to be cut off @saltychickk, so I can't really help you with that part!
This was an excellent response. Very detailed and in a logical order. Certainly worthy of a medal.