🔣 Inequality Symbols — Know These COLD!
- > greater than | < less than | ≥ greater than OR EQUAL TO | ≤ less than OR EQUAL TO
- OPEN circle ○ = NOT included — use with > or < CLOSED circle ● = IS included — use with ≥ or ≤
- KEY RULE: When you MULTIPLY or DIVIDE by a NEGATIVE number → FLIP the inequality sign!
📍 Graphing on Number Line
- x > 3: open ○ at 3, arrow →
- x < 3: open ○ at 3, arrow ←
- x ≥ 3: CLOSED ● at 3, arrow →
- x ≤ 3: CLOSED ● at 3, arrow ←
- 2 < x ≤ 5: open at 2, closed at 5
- shade between them!
🔧 Solving 1-Step Inequality
- Treat like an equation EXCEPT:
- If multiply/divide by NEGATIVE
- → FLIP THE SIGN!
- e.g. −2x < 8
- Divide by −2, flip sign:
- x > −4 (sign flipped!)
🔧 Solving Multi-Step
- e.g. 3x + 2 ≥ 11
- Step 1: subtract 2: 3x ≥ 9
- Step 2: divide by 3: x ≥ 3
- Graph: closed ● at 3
- shade to the right →
- Check: x=4 → 14≥11 ✓
📐 Two-Variable Inequality
- y > 2x+1: dashed line, ABOVE
- y < 2x+1: dashed line, BELOW
- y ≥ 2x+1: SOLID line, above
- CHECK: plug (0,0) in inequality
- If true → shade that side
- If false → shade other side!
x > 2
x ≤ 4
−1 ≤ x < 3
🔀 System of Two Inequalities → Find the OVERLAP Region
- Graph BOTH inequalities on the same coordinate plane. Solution = where the SHADING OVERLAPS!
- e.g. y > x+1 AND y ≤ −x+4: Graph first (dashed, shade above). Graph second (solid, shade below).
- OVERLAP region = solution set. Pick any point in overlap & plug into BOTH inequalities to verify!
🔢 Desmos Strategies
3 Ways to Get the Answer!
Strategy 1: Evaluate / Find a Value
- Type with CAPITAL letters
- e.g. 3*X + 2 >= 11
- Add slider for X
- X=3 → 11≥11 TRUE (boundary)
- X=4 → 14≥11 TRUE (solution!)
- X=2 → 8≥11 FALSE (not solution)
Strategy 2: Graph / Visualize
- Type y > 2*x + 1
- Desmos SHADES the region!
- Type y <= -x + 4
- DARKER overlap = solution!
- Click any point in overlap
- to verify it's a solution!
Strategy 3: Generate Table
- Type 3*x + 2 >= 11 → Table
- Find which x-values make TRUE
- Those x-values = solution set!
- Also: type x > 3
- Desmos shows number line!
- Shades correct side for you!
Remember This!
- FLIP THE SIGN when multiplying or dividing by a NEGATIVE! e.g. −3x > 9 → x < −3 (FLIPPED!)
- Dashed line = NOT included (> or <). Solid line = IS included (≥ or ≤).
- CHECK your answer: pick a point in shaded region, plug into the inequality — it MUST be TRUE!