Heart of Algebra

Linear Inequalities

Flip the sign, find the shade. Master the visual logic of SAT inequalities.

Boss Mode
Mastery
Heart of Algebra

The Negative Flip

\[ -2x < 10 \implies x> -5 \]

Multiplying or dividing by a negative flips the symbol.

Practice: "Solve for \(x\): \( -3x + 4 \geq 19 \)"

Show Solution & Analysis
🚫 School Way (Rigorous)

1. Subtract 4:
\( -3x \geq 15 \)
2. Divide by -3:
Flip symbol: \( x \leq -5 \)
Result: \( x \leq -5 \)

✅ Practix Way (Optimal)

**Step 1:** Add \( 3x \) to the other side to make it positive.
\( 4 \geq 3x + 19 \implies -15 \geq 3x \).
**Step 2:** Divide by positive 3.
\( -5 \geq x \).
**Result: \( x \leq -5 \)**
Moving the variable prevents "forgetting the flip".

Visual
Mastery
Heart of Algebra

Inequality Shading

\[ y > \text{ Above}, \quad y < \text{ Below} \]

Isolate \(y\) first. Use solid lines for \(\leq, \geq\) and dashed for \(<,>\).

Practice: "For \( y \leq -2x + 4 \), where is the shade?"