๐ Understanding Linear Equations
A linear equation represents a straight line on a graph. It shows a relationship where each value of $x$ corresponds to exactly one value of $y$. Think of it as a precise path!
๐ Definition: An equation that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
- ๐ Graphical Representation: A straight line. Every point on the line is a solution to the equation.
- ๐งฎ Solution: The set of all points $(x, y)$ that satisfy the equation. These points lie directly on the line.
๐ก Understanding Linear Inequalities
A linear inequality, on the other hand, represents a region on the graph. Instead of a precise line, it shows a range of possible values. It's like a zone of solutions!
๐งช Definition: An inequality that can be written in the form $y > mx + b$, $y < mx + b$, $y \geq mx + b$, or $y \leq mx + b$.
- ๐บ๏ธ Graphical Representation: A region of the coordinate plane bounded by a line. The region includes all points that satisfy the inequality.
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Solution: The set of all points $(x, y)$ that satisfy the inequality. These points lie in the shaded region.
๐ Comparison Table: Linear Equations vs. Linear Inequalities
| Feature |
Linear Equations |
Linear Inequalities |
| Definition |
$y = mx + b$ |
$y > mx + b$, $y < mx + b$, $y \geq mx + b$, or $y \leq mx + b$ |
| Graph |
A straight line |
A shaded region bounded by a line |
| Solution |
Points on the line |
Points in the shaded region |
| Boundary Line |
Solid line |
Solid line (for $\geq$ or $\leq$), Dashed line (for $>$ or $<$) |
๐ Key Takeaways
- ๐ฏ Equations show equality: Linear equations represent a precise relationship, shown as a line.
- ๐งญ Inequalities show a range: Linear inequalities show a range of possible values, shown as a shaded region.
- โ๏ธ Boundary lines matter: The type of line (solid or dashed) in an inequality indicates whether the points on the line are included in the solution.