Systems of Inequalities

What is a System of Inequalities?

A system of inequalities consists of two or more inequalities that must be satisfied simultaneously. On the SAT, you'll graph these to find the solution region.

Graphing Steps

Example: Graph the system:

y > 2x + 1 \\ y \leq -x + 4 \end{cases}$$ ### Step 1: Graph Each Inequality **First inequality:** $y > 2x + 1$ - Graph the line $y = 2x + 1$ (dashed line since it's $>$, not $\geq$) - Shade above the line **Second inequality:** $y \leq -x + 4$ - Graph the line $y = -x + 4$ (solid line since it's $\leq$) - Shade below the line ### Step 2: Find the Overlap The **solution** is where the shaded regions overlap. ## SAT Question Types ### Type 1: Identify the System from a Graph **Strategy:** - Check whether lines are solid ($\leq$ or $\geq$) or dashed ($<$ or $>$) - Test a point in the shaded region to verify the inequality direction ### Type 2: Determine if a Point is in the Solution Region **Test point $(2, 3)$ in the system above:** - $3 > 2(2) + 1 \rightarrow 3 > 5$ ✗ (FALSE) - Since it fails one inequality, $(2, 3)$ is NOT in the solution ### Type 3: Word Problems **Example:** A store sells notebooks ($x$) and pens ($y$). - You need at least 5 items total: $x + y \geq 5$ - You can spend at most $20: $2x + 3y \leq 20$ ## Quick Tips ✓ **Solid vs Dashed:** $\leq$ and $\geq$ use solid lines; $<$ and $>$ use dashed ✓ **Shade Direction:** Use test point $(0, 0)$ if not on the line ✓ **Solution Region:** Look for the overlap of all shaded areas ✓ **Boundary Check:** Points ON dashed lines are NOT solutions ## Common SAT Mistakes ❌ Confusing solid and dashed lines ❌ Shading the wrong side ❌ Not checking if points are in BOTH regions ❌ Forgetting that dashed lines exclude boundary points