"and" problem -1 = -3 + 2x < 17 "or" 7 – x = 6 or 7x – 1 > 27

Question

One-Variable Compound Inequalities
Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing.
• Compound inequalities
• And
• Or
• Intersection
• Union
Solve the compound inequalities as demonstrated in Elementary and Intermediate Algebra: 1. Be careful of how a negative x-term is handled in the solving process. 2. Show the solution sets written algebraically and as a union or intersection of intervals. 3. Describe in words what the solution sets mean, and then display a simple line graph for each solution set.
This is my “and” and “or” compound inequality problems:
If the first letter of your last name is Your “and” compound inequality is Your “or” compound inequality is
H -1 ≤ -3 + 2x < 17 7 – x ≥ 6 or 7x – 1 > 27

1. My “and” compound inequality problem: -1 ≤ -3 + 2x < 17.
What that mean is the inequality must….?



2. My “or” compound inequality problem: 7 – x ≥ 6 or 7x – 1 > 27.
What this means is that there are…..?

Answer 1

We use compound inequalities when we want to find more than two sets of a solution that have a certain range.  These inequalities consist of 2 or more comparison symbols such as <, >, ≤,  and ≥.

 

When we solve compound inequalities, the solution set is usually written with the terms or and and.   We use the term or when the solution sets have a gap with the ranges.  An example of an or compound inequality is

 

x < 5   or   x > 8

 

This means that the values between 5 and 8 are NOT included in the solution set.

 

We use the term and when solution set is in between the ranges.  Here is an example.

 

x > 5  and  x < 8

 

This means that the values in between 5 and 8 are included in the solution set.  We can also write the solution of the and inequality as

 

5 < x < 8

 

If  5 < x, then  x > 5.   x < 8 remains.  The logic is not altered.

 

 

We use the term union (∪)  to write the solution set in interval notation.  Take  5 < x < 8 for example.  In interval notation we write it as

 

(5, 8)

 

 

In interval notation,    x < 5   or   x > 8     is

 

(-∞, 5)∪(8, ∞).

 

When we have an intersection, this means that the inequality share common values within the set.  You can think of it like this:

 

and is to intersection

or is to union

 

 

Now we will solve each inequality.

 

1)

 

-1 ≤ -3 + 2x < 17.

 

The goal here is to solve for x, but x will not just have one value.  It will have a range of values.  But we still solve it the same way like an equation.

 

 

We add 3 on all sides of the inequality.  This will isolate the 2x in the middle.

 

2 ≤ 2x < 20

 

 

Next, divide all sides of the inequality by 2.  This will give use x in a range of values.

 

1 ≤ x < 10

 

 

This inequality means that the solution set is all values of x between 1 and 10, including 1 and excluding 10.

 

 

2)

 

 

7 – x ≥ 6     or      7x – 1 > 27

 

Since we have an or term here, we can see that we will have a gap in the solution set.  What we do here is solve the inequalities separately.  Solve for x, just as we did in the first problem.

 

7 – x ≥ 6                 or             7x – 1 > 27

   - x ≥ -1                or             7x > 28

     x ≥ 1                  or              x > 4

 

One thing to know is that when we divide by a negative number in an inequality, we flip the sign.  This is because we must obey the law of signs when we multiply and divide negative and positive numbers.  Recall that 2 negatives give a positive, a negative and a positive gives a negative.

 

x ≥ 1           or          x > 4

 

Now we compare these in equalities.  Both of them indicate that x is greater than 1.   Therefore, we chose the solution set

 

x > 4

 

This is because if x is greater than 4, it is already greater than 1.

 

The reason why we do NOT chose x ≥ 1  is because the values 2, 3, and 4 are NOT greater than 4, although they are greater than 1.