In this Lesson we will be doing more work with compound inequalities.
Graph these “or” statements on a number line.
x ≤ 2 or x > 5
Just to remind ourselves what we are doing, we are plotting the solution set of values to the statement x ≤ 2 or x > 5 on a number line, the lines represent all the values of x that satisfy the statement.
We will have two lines, one starting at 2 and one starting at 5,
Write a solid dot over 2 and a hollow dot over 5, with arrowed lines extending over the values that are less than or equal to 2 or greater than 5,
x > 8 or x < -1
Write a hollow dot over -1 and 8, then an arrowed line extending over all the values that are either less than -1 or greater than 8
Graph these “and” statements on number line, then look at the graphed number lines to help you write one single inequality.
x > -3 and x < 3,
Before we write our number line, remember what the statement means, x > -3 and x < 3,
The solution set that we graph will be the values that satisfy both parts in this statement, as it is an “and” statement
So, the solution set is all the values that are greater than -3 and less than 3, first lets try graphing x > -3 and x < 3
The solution set to the statement x > -3 and x < 3 is the values that satisfy both parts, visually this is the part of the graph that is covered by both lines that we plotted in the line graph above.
Now, to properly graph the solution set on a line graph for x > -3 and x < 3, we write the line that is overlapped by the line we drew for x > - 3 and the line we drew for x < 3,
Now we can use this visual to help us write a singular inequality,
We want to write a singular statement that says x is greater than - 3 and less than 3,
This will look like -3 < x < 3,
Compare this to the line we have written, and you may be able to see the similarities.
x ≥ 1 and x ≤ 6
Again, to help us write the solution set of the statement, lets first write out an individual line for x ≥ 1 and then one for x ≤,
Then when we have graphed those two lines, graph the line that will be the solution set to the statement above that (it will be the line where the two other lines have overlapped),
Now, we want to write a singular inequality that represents the values greater than or equal to 1 and less than or equal to 6,
Similar to the previous question, except this time we have ≤ / ≥ rather than < / >,
1 ≤ x ≤ 6
Now, we are going to practice doing the opposite of what we have just done.
When we talk about Singular Inequalities, note that Inequalities that involve “and” are almost always written as Singular Inequalities.
Graph the singular inequality, then rewrite the singular inequality as two inequalities
-1 ≤ x < 6
For graphing this, start with the two dots, a solid one above -1 and a hollow one above the 6, we then graph a line above all the values in between that,
Now we want to rewrite the singular inequality as two inequalities, it can help if we think of the singular inequality in words first, like x is greater than or equal to -1 and is less than 6, now write,
-1 ≤ x and x <6
You could also write it like,
x ≥ -1 and x < 6
2 ≤ x ≤ 5
First, we graph this inequality, starting with our dots, then graphing our line over the values in between those dots,
Now, we split this singular inequality into two inequalities,
2 ≤ x ≤ 5,
Writing this in words can help us translate it into two inequalities,
x is greater than or equal to 2 and x is less than or equal to 5,
x ≥ 2 and x ≤ 5,
Can also be written as
2 ≤ x and x ≤ 5,
Another way to think of this, is when you see a singular inequality, add in “and x” into the inequality,
So, 2 ≤ x ≤ 5
Becomes, 2 ≤ x and x ≤ 5, which is,
This might make it easier to see the inequality as two inequalities.
Look at the following line graphs, with solution sets graphed on them, and then determine what the compound inequality is.
Now we are doing the opposite of what we have been doing. We are seeing a line graph with a solution set for an inequality graphed onto it, then we determine what the compound inequality is based on that line graph.
Start by writing down what you see from the graph, first of all,
We have one line with two connected dots, this hints that we will be dealing with an “and” inequality,
If we want to break down the steps to finding our inequality, we can outline our inequality starting with,
Something and something
Now we see our two dots, note that one dot is over -2 and the other is over 3, we can now fill this part into our inequality outline,
Something with -2 and something with 3,
Now, we look at the dots and line,
The dot over -2 has its line extending to the right on the number line, this means our x values for this set will be either greater than -2 or greater than or equal to -2,
We look at the dot over the -2 and see it is hollow, which means the x values for this inequality will be greater than -2.
So, we now know one part of our inequality, our x values will be greater than -2, which looks like,
x > -2
The dot over the 3 has its line extending to the left on the number line, this means our x values for this set will be either less than 3 or less than or equal to 3.
We look at the dot over the 3 and see it is solid, which means the x values for this inequality will be less than or equal to 3.
So, we now know the other part of our inequality, our x values will be less than or equal to 3, which looks like,
x ≤ 3,
Now we combine these two statements, remembering it is an “and’ inequality, as its the line in between -2 and 3,
x > -2 and x ≥3
Now we have identified our two-part inequality, we can also write an equally correct version, that is more common with “and” inequalities,
-2 < x ≤ 3
First, we see two lines on the line graph going away from each other, this means that we will be dealing with an “or” inequality,
“something“ or “something”
We see we have our two dots over the values -1 and 4,
When we look at the lines extending away from these two values,
We see that the line extending away from -1 is going to the left, this means that our x values for this inequality will be less than or less than or equal to -1.
When we look at the dot over the -1 we see it is a solid dot, which means we include the -1 value in our set, which means we will have our x values less than or equal to -1, which looks like,
x ≤ -1
We see that the line extending away from the 4 is going to the right, this means that our x values for this inequality will also be greater than or greater than or equal to 4.
When we look at the dot over the 4 we see it is a hollow dot, this means we will not include the 4 value in our set, which means we will have our x values also greater than 4, which looks like,
Combining all the skills we have been developing, solve the given inequality and graph its solution set on a number line.
2x + 3 ≥ 7 and -2x + 6 > -10
Here we are combining all the skills we have developed in this Unit, we are asked to graph this inequality, but in order to do that, we must know the solution set, which means we have to solve it!
Let’s first solve each part of the inequality, individually,
Let’s solve 2x + 3 >= 7
Remember we can solve inequalities almost the exact same way we solve equations, except when we multiply or divide both sides by a negative. When we do that, we have to switch the inequality (i.e. switch < to > or ≤ to ≥)
So, we start by subtracting both sides by 3,
2x + 3 - 3 ≥ 7 - 3
2x ≥ 4,
Now, divide both sides by 2, (not dividing by negative, so don't have to switch the inequality)
2x / 2 ≥ 4 / 2
x ≥ 2,
The we have our first part of the inequality, we now know, part of our solution set will be the values that are greater than or equal to 2,
Now solve, -2x + 6 > -10,
Any time we are solving an inequality and see a negative multiplying our x, take a note that we will most likely have to divide by a negative at some point when solving, and have to switch the inequality.
First, subtract both sides by 6,
-2x + 6 - 6 > -10 - 6
-2x > -16
Now we divide both sides by -2, and switch the inequality,
-2x / -2 < -16 / -2
x < 8,
There we have our two parts of the inequality,
x > 2 and x < 8,
Or, as this is an “and” inequality, we can more commonly write it as,
2 < x < 8,
Now we can graph this,
First, graph our line for,
x > 2
A hollow dot over 2,
Our x values are greater than 2, so a line extending to the right of 2
Now on that same line graph,
Graph the line for,
x < 8
A hollow dot over 8,
Our x values are less than 8, so a line extending to the left of 8,
Now we see our two lines, this is an “and” inequality, so we want a line that is true for both parts of the inequality,
So, we are looking the line where these two parts overlap,
This will look like,
½ (x + 2) < 4 or -3(x - 3) ≤ 12
First, note we have an “or” inequality, which refreshing our definition means,
A value is a solution the inequality if it makes at least one parts of the inequality true,
Let’s solve each part individually,
½ (x + 2) < 4,
Multiply both sides 2 (the same as dividing both sides by a half),
(½ (x + 2))2 < (4)2,
(x + 2) < 8,
Subtract both sides by 2,
x + 2 - 2 < 8 - 2
x < 6,
Now for the other part,
-3(x - 3) ≤ 12,
Divide both sides by -3, (remember, switch the inequality, as we are dividing by a negative)
(-3(x - 3)) / -3 ≥ (12) / -3,
x - 3 ≥ -4,
Add 3 to both sides,
x - 3 + 3 ≥ -4 + 3
x ≥ -1
So, we have our two parts to the “or” inequality,
x < 6 or x ≥ -1,
Now we graph this,
Let’s graph the two parts on one line graph,
Note something interesting,
As this is an “or” inequality, we aren't looking for the values that have to satisfy both parts of the inequality, we are looking to graph the values that can satisfy one part or the other part or both,
Try and think of a number that is not either less than 6 or greater than or equal to -1,
There is not one!
This means that every number will satisfy this inequality!
Solve 3x + 1 ≤ 13 or -2x - 3 < 3, State the solutions.
Firstly, note we have an or statement, so for our solution, we still need to solve each inequality, but just write or between the two x-value inequality solutions.
Start with 3x + 1 ≤ 13
Subtract 1 from both sides,
3x + 1 - 1 ≤ 13 - 1
3x ≤ 12
Divide both sides by 3,
3x / 3 ≤ 12 / 3
x ≤ 4,
For the other, -2x - 3 < 3
Add, 3 to both sides,
-2x - 3 + 3 < 3 + 3
-2x < 6
Divide both sides by -2, remember to switch the inequality!
-2x / -2 > 6 / -2
x > -3
Complete Solution: x > -3 or x ≤ -4