- #1
WannaBe
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Express the set {X E R: (x+3) (7-x) ((x-2)^2) > 0} as a union of intervals
Express the set as a union of intervals:
In summary, the solution set for the inequality {X E R: (x+4) (x-1)(x-5)^2(x-12)^3>0} is the interval (-4,1)\,\cup\,(12,\infty).
- #2
alyafey22
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What have you tried ? how to solve that inequality ?
- #3
WannaBe
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ZaidAlyafey said:
(x+3) (7-x) ((x-2)^2) > 0}
x > -3 , x < 7 , x > 2
(-3,2)U(7,oo)
Is it correct?
- #4
alyafey22
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Remember that
\(\displaystyle ab>0 \,\,\, \text{iff }\,\,\,\, a>0,b>0 \,\,\, \text{or}\,\,\, a<0,b<0\)
\(\displaystyle ab>0 \,\,\, \text{iff }\,\,\,\, a>0,b>0 \,\,\, \text{or}\,\,\, a<0,b<0\)
- #5
WannaBe
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ZaidAlyafey said:
So, what's the next step?
- #6
alyafey22
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Note that if you use numbers greater than 7 the inequality will not hold.
- #7
MarkFL
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I have moved this topic. Although this question does involve sets, rewriting a solution set for an inequality given in set-builder notation to interval notation is a topic typically studied by students of "elementary" algebra.
I don't want to trample on the help being given by ZaidAlyafey, so I will walk you through a similar problem, the way I was taught.
Suppose we are given the set:
\(\displaystyle \{x|(x+4)(x-1)(x-5)^2(x-12)^3>0\}\)
Step 1: Draw a real number line and mark the roots of the polynomial expression:
View attachment 1434
Step 2: Consider whether the inequality is weak or strict. If weak, put solid dots at the roots to show they are part of the solution set (giving closed intervals), and if strict put hollow dots to indicate they are not part of the solution (giving open intervals). For this problem, we have a strict inequality so we will put hollow dots at the roots:
View attachment 1435
Step 3: Choose test values from within each interval into which the roots have divided the real number line. I will choose some and put them in red, however the choice of values is up to the person working the problem, as long as they are within the intervals:
View attachment 1436
Step 4: Put each test value into each factor of the polynomial, and record the resulting signs of each factor, and the consider what the sign the resulting product must be. An even number of negatives gives a positive while an odd number of negatives gives a negative. Take care to make sure the factors with exponents are counted the correct number of times:
View attachment 1437
As you become more proficient at this step, you will see that when a root has an odd multiplicity, the sign of the polynomial will change across the root, and when the root has an even multiplicity it will not. The multiplicity of a root refers to how many times it occurs, as indicated by its exponent. Notice the root $x=5$ is of multiplicity 2, and the sign of the polynomial did not change, whereas the root $x=12$ is of multiplicity 3 and the sign did change, as it did also for the other roots of multiplicity 1. Using this information, it is then really only necessary to check one interval, and then apply the information regarding the multiplicity of each root accordingly.
Step 5: Since we are interested in those intervals where the polynomial is positive, we then shade those intervals which resulted in a positive sign:
View attachment 1438
Step 6: Write the solution in interval notation:
\(\displaystyle (-4,1)\,\cup\,(12,\infty)\)
I don't want to trample on the help being given by ZaidAlyafey, so I will walk you through a similar problem, the way I was taught.
Suppose we are given the set:
\(\displaystyle \{x|(x+4)(x-1)(x-5)^2(x-12)^3>0\}\)
Step 1: Draw a real number line and mark the roots of the polynomial expression:
View attachment 1434
Step 2: Consider whether the inequality is weak or strict. If weak, put solid dots at the roots to show they are part of the solution set (giving closed intervals), and if strict put hollow dots to indicate they are not part of the solution (giving open intervals). For this problem, we have a strict inequality so we will put hollow dots at the roots:
View attachment 1435
Step 3: Choose test values from within each interval into which the roots have divided the real number line. I will choose some and put them in red, however the choice of values is up to the person working the problem, as long as they are within the intervals:
View attachment 1436
Step 4: Put each test value into each factor of the polynomial, and record the resulting signs of each factor, and the consider what the sign the resulting product must be. An even number of negatives gives a positive while an odd number of negatives gives a negative. Take care to make sure the factors with exponents are counted the correct number of times:
View attachment 1437
As you become more proficient at this step, you will see that when a root has an odd multiplicity, the sign of the polynomial will change across the root, and when the root has an even multiplicity it will not. The multiplicity of a root refers to how many times it occurs, as indicated by its exponent. Notice the root $x=5$ is of multiplicity 2, and the sign of the polynomial did not change, whereas the root $x=12$ is of multiplicity 3 and the sign did change, as it did also for the other roots of multiplicity 1. Using this information, it is then really only necessary to check one interval, and then apply the information regarding the multiplicity of each root accordingly.
Step 5: Since we are interested in those intervals where the polynomial is positive, we then shade those intervals which resulted in a positive sign:
View attachment 1438
Step 6: Write the solution in interval notation:
\(\displaystyle (-4,1)\,\cup\,(12,\infty)\)
Attachments
Related to Express the set as a union of intervals:
1. What does it mean to "express a set as a union of intervals"?
When expressing a set as a union of intervals, we are essentially finding a way to represent the set as a combination of other sets or intervals. This can be useful in situations where the original set is complex or difficult to work with, and expressing it as a union of intervals can make it easier to understand or manipulate.
2. How do you express a set as a union of intervals?
To express a set as a union of intervals, we first need to determine the intervals that make up the set. This can be done by examining the elements of the set and identifying any patterns or gaps. Then, we can combine these intervals using the union operator (represented by the symbol ∪) to create a single set that encompasses all the elements of the original set.
3. Can a set be expressed as a union of intervals in more than one way?
Yes, a set can often be expressed as a union of intervals in multiple ways. This is because there are often different intervals that can be combined to create the same set. However, some sets may have a unique way of being expressed as a union of intervals.
4. Why would you want to express a set as a union of intervals?
Expressing a set as a union of intervals can be useful in a variety of situations. It can make the set easier to understand or manipulate, as well as provide a more concise representation of the set. This can be especially helpful in mathematical or scientific contexts where precise and efficient notation is important.
5. Are there any limitations to expressing a set as a union of intervals?
While expressing a set as a union of intervals can be helpful in many cases, there are some limitations. For example, not all sets can be easily expressed as a union of intervals, especially if they are very complex or have irregular patterns. Additionally, this method of representation may not always be the most efficient or practical, depending on the specific context or purpose.
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