7C Inequalities & Graphing Study Guide

Key Vocabulary

Inequality — a math sentence that compares two values using <, >, ≤, or ≥ instead of =. (desigualdad)

Greater than (>) — the value on the left is larger than the value on the right. (mayor que)

Less than (<) — the value on the left is smaller than the value on the right. (menor que)

At least (≥) — greater than or equal to; the minimum allowed value. (al menos)

At most (≤) — less than or equal to; the maximum allowed value. (como máximo)

Open circle — used on a number line for < or > (value is NOT included). (círculo abierto)

Closed circle — used on a number line for ≤ or ≥ (value IS included). (círculo cerrado)

Number line — a line with numbers in order, used to show solutions of inequalities. (recta numérica)

Visual Example — Number Line Graphs

x > 3    (x is greater than 3)
  ←-----|-----|---(o)===>
         1     2     3     4     5
  Open circle at 3, shade RIGHT
  (3 is NOT included)

x ≤ 5    (x is less than or equal to 5)
  ←==================(•)-----|--->
         2     3     4     5     6
  Closed circle at 5, shade LEFT
  (5 IS included)

Step-by-Step: Write & Graph Inequalities

1 Read the problem — Find the key words (at least, at most, more than, fewer than, no more than).
2 Choose the symbol:
• "more than" or "greater than" → >
• "fewer than" or "less than" → <
• "at least" or "no fewer than" → ≥
• "at most" or "no more than" → ≤
3 Write the inequality using a variable for the unknown.
4 Graph it:
• Draw a number line.
• Put an open circle for < or >.
• Put a closed circle for ≤ or ≥.
• Shade in the direction of the solutions.

Symbol Reference

< less than | > greater than

≤ less than or equal | ≥ greater than or equal

Watch Out!

Open circle = NOT included (> or <). Closed circle = IS included (≥ or ≤).
Direction matters! > and ≥ shade to the RIGHT. < and ≤ shade to the LEFT.
"At least 5" means 5 or more (x ≥ 5), not x > 5.
An inequality has MANY solutions, not just one. For example, x > 3 includes 4, 5, 6, 3.5, 100, etc.

Practice Problems

1. You must be at least 48 inches tall to ride a roller coaster. Which symbol goes in the blank?
h ___ 48. Type: >, <, >=, or <=

2. A bag can hold at most 15 pounds. Write the inequality for weight w. What symbol do you use?
w ___ 15. Type: >, <, >=, or <=

3. For the inequality x > 7, what type of circle do you draw at 7? Type "open" or "closed".