Integrated Math 2–3 Honors · Lesson Review · August 17, 2026

Interval Notation and Graphing Inequalities

Read and write intervals, solve a one-variable inequality, and use a boundary line plus a test point to graph a two-variable solution set.

22-minute edited lesson8 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Intervals and Interval Notation 4:0402Open, Closed, and Half-Open Intervals 7:2003Intervals with Infinity 9:4604Intervals Versus Ordered Pairs 10:5805Solving a One-Variable Inequality 13:4306Graphing a Two-Variable Inequality 17:3707Half-Planes and Dashed Boundaries 20:0408A False Test Point and Opposite-Side Shading

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02 · Understand

Learning targets

Translate between number-line graphs, inequalities, and interval notation.
Use square brackets and round parentheses to include or exclude endpoints.
Explain why positive and negative infinity always use parentheses.
Solve a one-variable inequality and reverse its sign when dividing by a negative.
Graph a two-variable inequality with the correct boundary and shaded half-plane.

03 · Vocabulary

Key definitions

Use these terms precisely when you explain a graph or solution set.

01Interval: A continuous set of real numbers between two endpoints; each endpoint may be included or excluded.
02Endpoint: A boundary value at the left or right end of an interval.
03Closed endpoint: An included endpoint, shown with a filled dot on a number line and a square bracket in interval notation.
04Open endpoint: An excluded endpoint, shown with an open dot on a number line and a round bracket in interval notation.
05Half-open interval: An interval with one endpoint included and the other excluded; it is also called half-closed.
06Interval notation: A compact way to name an interval by listing its left and right boundaries with brackets or parentheses.
07Solution set: The set of every value or ordered pair that makes an equation or inequality true.
08Boundary line: For a two-variable inequality, the related equation whose graph separates the two half-planes.
09Test point: A point not on the boundary used to decide which side of a boundary line satisfies an inequality.
10Half-plane: One of the two regions made by a line in the coordinate plane.

04 · Apply

Worked examples

Match each symbol to what is included in the solution set.

Read intervals

Included and excluded endpoints

Notation	Meaning
$[2,5]$	From 2 through 5, including both endpoints
$(1,4)$	Between 1 and 4, excluding both endpoints
$[-5,10)$	Include −5; exclude 10
$(0,\pi]$	Exclude 0; include π

Use infinity

One infinite direction

\begin{aligned}x\ge4&\;\Longleftrightarrow\;[4,\infty)\\x<3&\;\Longleftrightarrow\;(-\infty,3)\end{aligned}

Infinity receives a parenthesis because it is not a real-number endpoint.

Every real number

(-\infty,\infty)=\mathbb{R}

Both directions continue forever; neither infinity symbol is included.

One variable

Solve $-2x+5\le-11$

$-2x+5\le-11$
$-2x\le-16$ subtract 5 from both sides
$x\ge8$ divide by −2 and reverse the inequality

Solution set in interval notation: $[8,\infty)$ .

Two variables

Graph $y\le3x+2$

$y=3x+2$ graph the boundary line
$(0,0):\;0\le3(0)+2$ true test point
$0\le2$ shade the side containing (0, 0)

The boundary is solid because points on the line satisfy the “or equal to” part.

Strict inequality

Graph $y<-3x-2$

$y=-3x-2$ draw a dashed boundary because the inequality is strict
$(0,0):\;0<-3(0)-2$
$0<-2$ false, so shade the side opposite (0, 0)

05 · Avoid

Common mistakes

Brackets do not mean “left” or “right.”

Square means included; round means excluded, on either side.

Never include infinity.

Write $[4,\infty)$ , not $[4,\infty]$ .

Reverse after dividing by a negative.

From $-2x\le-16$ , the result is $x\ge8$ .

Match the boundary to the symbol.

Use a solid line for ≤ or ≥ and a dashed line for < or >.

A test point chooses a side.

A false test means shade the half-plane opposite the test point.

Interval notation is one-dimensional.

It describes sets of numbers, not regions made of ordered pairs.

06 · Check yourself

Quick check

Decide first, then open the answer.

1Write “between −3 and 6, including −3 but not 6” in interval notation.

$[-3,6)$ . The square bracket includes −3; the parenthesis excludes 6.

2Why is the infinity endpoint in

[4,\infty)

round?

Infinity is not a real number. It describes continuing forever, not an endpoint that can be included.

3Solve

-3x>12

$x<-4$ . Dividing by −3 reverses the inequality. Interval notation: $(-\infty,-4)$ .

4Should the boundary for

y\ge2x-1

be solid or dashed?

Solid. The equality part means points on $y=2x-1$ are solutions.

5If a test point makes a two-variable inequality false, which side is shaded?

The opposite side of the boundary. The half-plane containing the false test point is not the solution region.

07 · Revisit

Transcript and vocabulary practice

Search the corrected, timestamped transcript for a term or example. Then use the vocabulary practice connected to the eKadence activity to write each definition and create your own example.

Download corrected transcript