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Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 1 of 17
Linear and Compound Inequalities
In the past we have solved inequalities that have a solution set.
For example, let’s solve:
8 − (4𝑥 − 2) ≥ −5(𝑥 + 1) − 5
At this point, we have a choice to either graph our solution, write the solution using interval notation, or we can
write the solution using set-builder notation.
Graph:
Interval Notation:
Set-builder Notation:
This solution means that the variable x can be any real value that is 20 or larger to make the inequality
TRUE!!
Notes about interval notation:
When you write the interval notation that does NOT include the endpoint, use parenthesis.
Always use a parenthesis when you use a ∞ or −∞.
When using interval notation, is always listed first (e.g. , 4 ) and is always listed second (e.g.
4, ).
Remember when you divide or multiply by a negative number, you need to the inequality.
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 2 of 17
Objective 1: Solving Linear Inequalities with No Solution and All Real Numbers
Sometimes you may run across an inequality that has no solution or all real numbers. Let’s see what this may
look like.
Example: Solve the inequality, graph and write the solution using interval and set-builder notation.
−29 − 42𝑥 > 6(1 − 7𝑥)
As you may notice, 29 is NOT greater than 6. This inequality is false and so we would say that it has “No
Solution”.
The graph of a “No Solution” answer would be just a plain number line.
To write the solution using interval or set-builder notation we use the “empty set” symbol:
What will happen if we switch the inequality around in the same exact problem from before?
−29 − 42𝑥 < 6(1 − 7𝑥)
We end up with a TRUE inequality. 29 is less than 6 which means no matter what value we give the variable
x, the inequality will always be true.
Here is how we would write the solution in all 3 forms:
Graph:
Interval Notation: (−∞, ∞)
Set-builder Notation: {𝑥|𝑥 ∈ ℝ}
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 3 of 17
Example: Solve the inequality, graph and write the solution using interval and set-builder notation.
6 4 6 24x x
Example: Solve the inequality, graph and write the solution using interval and set-builder notation.
6 3 3 6x x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 4 of 17
Pause the video and try these problems.
Solve the inequality and write the solution using set-builder and interval notation.
1. 5 8 3 9 2
2. 2 4 3 8 5 7
3. 4 10 7 3 2
4. 6 3 2 4 5 17
x x x
x x x
x x x
x x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 5 of 17
Objective 2: Solving Compound Inequalities with “and”
Compound inequalities involve a connection between more than one inequality. First let's look at two
inequalities connected with the word "and". When we solve for a conjunction (“and”) we look to see where the
two sets intersect or overlap.
It is often helpful to use a graph to determine the solution.
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
5 6 2 3 2 4x x and x
Let’s now graph both of the solutions on the same number line.
The solution to this problem is where the two graphs overlap.
Graph:
Interval Notation:
Set-builder Notation:
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 6 of 17
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
3 8 6 7 9 2 4 10x x and x x x
Graph both of the solutions on the same number line.
Graph:
Interval Notation:
Set-builder Notation:
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 7 of 17
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
7𝑥 + 3(2𝑥 − 4) < 5(3𝑥 + 1) − 3 𝑎𝑛𝑑 5𝑥 − 10 > 2(2𝑥 − 4)
Graph both of the solutions on the same number line.
Graph:
Interval Notation:
Set-builder Notation:
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 8 of 17
At times, compound inequalities can be written like:
−6 < 6𝑥 + 12 ≤ 9
There are a couple of methods to solving compound inequalities like this.
Method 1:
The goal is to isolate the variable in the middle of the inequality. You need to remember that whichever
inverse operation you use to isolate the variable, you need to do that same operation to all sides of the
inequality.
Graph:
Interval Notation:
Set-builder Notation:
Method 2:
Break up the inequality by re-writing it as 2 different inequalities.
−6 < 6𝑥 + 12 ≤ 9
−6 < 6𝑥 + 12 𝑎𝑛𝑑 6𝑥 + 12 ≤ 9
From here you would solve each inequality and the solution would still be the overlapped portion of the
two graphs together.
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 9 of 17
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation
−10 < 4 − 2𝑥 ≤ −6
Graph:
Interval Notation:
Set-builder Notation:
Pause the video and try these problems.
For each problem, graph the solution and write the solution using interval and set-builder notation.
1. 8 4 5 16x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 10 of 17
2 1
2. 4 5 6 and 1 83 4
x x
3. 2 2 4 14x
1 4
4. 3 4 6 and 7 72 5
x x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 11 of 17
Objective 3: Solving Compound Inequalities with “or”
Next let's look at two inequalities combined with the word "or". Think of a family reUNION, where all are
invited from any of the family components. Anyone that is in any of the sets is part of the answer.
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
𝑥 < 3 𝑜𝑟 𝑥 > 7
Since 𝑥 is already by itself in both inequalities, let’s graph both of them on the same number line.
When two inequalities are combined with the word "or", it is the union of the two solution sets. So here
we want the set of numbers that are either less than 3 OR greater than 7.
Graph:
Interval Notation:
Set-builder Notation:
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
6 + 7𝑥 < 6𝑥 − 5 𝑜𝑟 3𝑥 − 7 ≤ 5 + 5𝑥
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 12 of 17
Let’s graph both of them on the same number line.
Graph:
Interval Notation:
Set-builder Notation:
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
2 5 7 8 5 07
xx x or
Let’s graph both of them on the same number line.
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 13 of 17
Graph:
Interval Notation:
Set-builder Notation:
Example: Solve, graph the solution set, and write the solution in interval and set-builder notation.
3 5 5 8x or x
Let’s graph both of them on the same number line.
Graph:
Interval Notation:
Set-builder Notation:
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 14 of 17
Pause the video and try these problems.
For each problem, graph the solution and write the solution using interval and set-builder notation.
1. 2 10 or 4 1 14x x x x
2. 3 7 1 4 2 or 8 2 3 6x x x
1
3. 6 2 2 3 1 4 or 8 2 3 3 14
x x x x x
1
4. 2 4 3 3 4 or 2 64
x x x x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 15 of 17
Objective 4: Solving Compound Inequalities with All Real Numbers and No Solution
Let us look at the following compound inequalities.
I. Solve: 5 and 7x x
Graph:
Interval Notation:
Set-builder Notation:
II. Solve: 2 or 4x x
Graph:
Interval Notation:
Set-builder Notation:
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 16 of 17
Summary of Compound Inequalities:
Inequalities AND vs. OR
𝑥 > 2 𝑥 > 5
Graph:
𝑥 > 2 𝑎𝑛𝑑 𝑥 > 5
Graph:
Interval Notation:
Set-builder Notation:
𝑥 > 2 𝑜𝑟 𝑥 > 5
Graph:
Interval Notation:
Set-builder Notation:
𝑥 ≤ 2 𝑥 > 5
Graph:
𝑥 ≤ 2 𝑎𝑛𝑑 𝑥 > 5
Graph:
Interval Notation:
Set-builder Notation:
𝑥 ≤ 2 𝑜𝑟 𝑥 > 5
Graph:
Interval Notation:
Set-builder Notation:
𝑥 > 2 𝑥 ≤ 5
Graph:
𝑥 > 2 𝑎𝑛𝑑 𝑥 ≤ 5
Graph:
Interval Notation:
Set-builder Notation:
𝑥 > 2 𝑜𝑟 𝑥 ≤ 5
Graph:
Interval Notation:
Set-builder Notation:
Pause the video and try these problems.
For each problem, write the solution using interval and set-builder notation.
1. 6 8 or 5 3 2 9x x x
Cypress College Math Department – CCMR Notes Linear and Compound Inequalities, Page 17 of 17
2. 9 12 3 and 2 8 10 2x x x
1 3
3. 4 3 5 and 2 65 4
x x
4. 4 1 3 7 and 2 4 3 8 1x x x x