1. Inequality Symbols
An inequality compares two quantities.
The four main inequality symbols are:
$$<$$
means less than
$$>$$
means greater than
$$\leq$$
means less than or equal to
$$\geq$$
means greater than or equal to
Examples:
$$x<5$$
means that \(x\) can be any real number smaller than 5.
$$x\geq2$$
means that \(x\) can be 2 or any number greater than 2.
2. Strict and Inclusive Inequalities
There are two important types of inequalities.
Strict inequalities
$$x<3$$
or
$$x>3$$
The endpoint is not included.
Inclusive inequalities
$$x\leq3$$
or
$$x\geq3$$
The endpoint is included.
This difference determines whether we use parentheses or square brackets in interval notation.
3. Open and Closed Endpoints
nterval notation uses:
Parentheses
$$(\ )$$
A parenthesis means the endpoint is not included.
For example:
$$(2,5)$$
means:
$$2<x<5$$
Neither 2 nor 5 is included.
Square brackets
$$[\ ]$$
A square bracket means the endpoint is included.
For example:
$$[2,5]$$
means:
$$2\leq x\leq5$$
Both 2 and 5 are included.
4. Four Basic Interval Types
Students need to recognize these immediately.
Open interval
$$(a,b)$$
means:
$$a<x<b$$
Neither endpoint is included.
Closed interval
$$[a,b]$$
means:
$$a\leq x\leq b$$
Both endpoints are included.
Left closed, right open
$$[a,b)$$
means:
$$a\leq x<b$$
Left open, right closed
$$(a,b]$$
means:
$$a<x\leq b$$
5. Inequalities Extending to Infinity
Some inequalities continue forever in one direction.
For example:
$$x>3$$
becomes:
$$(3,\infty)$$
Similarly:
$$x\geq3$$
becomes:
$$[3,\infty)$$
For negative infinity:
$$x<-2$$
becomes:
$$(-\infty,-2)$$
and:
$$x\leq-2$$
becomes:
$$(-\infty,-2]$$
Important rule
Infinity is never included as an endpoint.
Therefore, always use parentheses with:
$$\infty$$
and:
$$-\infty$$
Correct:
$$(-\infty,5]$$
Incorrect:
$$[-\infty,5]$$
Infinity is not a real number that can be included in the interval.
6. Converting Inequalities to Interval Notation
Example 1
$$x>4$$
becomes:
$$(4,\infty)$$
Example 2
$$x\leq7$$
becomes:
$$(-\infty,7]$$
Example 3
$$-2<x<5$$
becomes:
$$(-2,5)$$
Example 4
$$-3\leq x<4$$
becomes:
$$[-3,4)$$
The left endpoint is included, but the right endpoint is not.
7. Converting Interval Notation to Inequalities
You also need to work in the opposite direction.
Example 1
$$(1,6)$$
means:
$$1<x<6$$
Example 2
$$[-2,4]$$
means:
$$(-\infty,8]$$
Example 3
$$(3,\infty)$$
means:
$$x>3$$
Example 4
$$(-\infty,8]$$
means:
$$x\leq8$$
8. Number-Line Representation
I strongly recommend showing this visually in the video.
There are two endpoint symbols:
Open circle
An open circle means the endpoint is not included.
Use it for:
$$<$$
or
$$\geq$$
For:
$$x>2$$
show an open circle at 2 and shade to the right.
For:
$$x\leq2$$
show a filled circle at 2 and shade to the left.
This should definitely be animated in your video rather than explained only in text.
9. Compound Inequalities
A compound inequality places a variable between two boundaries.
For example:
$$2<x\leq7$$
means:
- \(x\) is greater than 2;
- \(x\) is less than or equal to 7.
The interval is:
$$(2,7]$$
Another example:
$$-4\leq x<3$$
becomes:
$$[-4,3)$$
10. “And” versus “Or”
This distinction becomes very important when solving inequalities.
AND
Consider:
$$x>1\text{ and }x<5$$
Both conditions must be true.
Therefore:
$$1<x<5$$
and the interval is:
$$(1,5)$$
OR
Consider:
$$x<-2\text{ or }x\geq3$$
There are two separate regions.
We write:
$$(-\infty,-2)\cup[3,\infty)$$
The symbol:
$$\cup$$
means union.
This connects directly with what you learned in Lesson 01.2.
11. Disconnected Solution Sets
Some inequality solutions consist of more than one interval.
For example:
$$x<-1\text{ or }x>4$$
cannot be written as one continuous interval.
Instead:
$$(-\infty,-1)\cup(4,\infty)$$
This structure will become extremely important when we study quadratic and rational inequalities.
12. All Real Numbers
If every real number satisfies a condition, the solution set can be written as:
$$(-\infty,\infty)$$
or:
$$x\in\mathbb{R}$$
13. No Solution
If no real value satisfies an inequality, the solution set is: