4-5 Additional Practice Systems of Linear Inequalities

A system of linear inequalities is a set of two or more linear inequalities that must be satisfied at the same time, and its solution is the region where all their shaded areas overlap. For “4-5 additional practice systems of linear inequalities,” the goal is to graph each boundary line, decide dashed versus solid, shade correctly, and then read the intersection region. A point is a solution only if it satisfies every inequality in the system, not just one.

Core Concepts Review

A linear inequality looks like y > 2x − 1 or 3x + 4y ≤ 12. The symbols matter: < and > mean the boundary line is not included, so it is drawn dashed. The symbols ≤ and ≥ mean the boundary is included, so it is drawn solid. Khan Academy notes that the graph of a single linear inequality is a half-plane — all the points on one side of the boundary line Khan Academy.

When two or more inequalities are combined, the solution set shrinks. Only points in the overlap satisfy the whole system. A quick test: plug the coordinates into every inequality. If even one fails, the point is out.

Take the system y ≥ x and y < 3. The point (2, 2) works: 2 ≥ 2 is true, and 2 < 3 is true. The point (0, 5) fails because 5 ≥ 0 is true but 5 < 3 is false. So (0, 5) is not a solution to the system, even though it satisfies the first inequality.

Purplemath stresses that checking a test point is the fastest way to confirm which side of a boundary to shade Purplemath. Use a point not on the line, such as (0, 0), whenever possible.

Graphing Solution Regions

Graphing a system follows a repeatable four-step routine.

1. Rewrite each inequality in slope-intercept form (y = mx + b) so you can plot the boundary line easily.
2. Draw each boundary line, using a dashed line for < or > and a solid line for ≤ or ≥.
3. Shade one side of each line. Pick a test point; if it satisfies the inequality, shade that side.
4. Identify the overlap. The darkest or clearly marked intersection is the solution region.

Each inequality alone describes an infinite half-plane. Two half-planes usually intersect in a wedge; three or more can form a polygon, a strip, or even no region at all. If the shaded areas never overlap, the system has no solution. Math is Fun points out that a system with no common region is called inconsistent Math is Fun. That is a legitimate answer, not a mistake.

A useful shortcut for shading: if the inequality is in y = mx + b form, “y >” or “y ≥” means shade above the line, and “y <” or “y ≤” means shade below. This works only after y is isolated.

Guided Example

Graph the system:

• y > x − 2
• y ≤ −x + 4

Step 1: The first boundary is y = x − 2, dashed because of >. The second is y = −x + 4, solid because of ≤.

Step 2: Plot both lines. The first passes through (0, −2) and (2, 0). The second passes through (0, 4) and (4, 0).

Step 3: Shade above the dashed line and below the solid line.

Step 4: The overlap is a wedge-shaped region that opens to the right and narrows as the lines converge. The point (1, 1) is inside the region and satisfies both inequalities.

A second check: the point (1, 1) gives 1 > 1 − 2, or 1 > −1, which is true, and 1 ≤ −1 + 4, or 1 ≤ 3, which is also true. If a test point satisfies all inequalities, it is inside the region. If it fails any one, it is outside. That single habit catches most graphing errors before they become pattern mistakes.

Parallel boundary lines deserve special attention. If a system pairs y > 2x + 1 with y < 2x − 3, both boundary lines have slope 2, so they never meet. The first shades above one line, the second shades below a parallel line that sits lower on the graph. The shaded regions never overlap, so the system has no solution. Recognizing parallel boundaries with opposing directions is a quick way to spot an inconsistent system.

Key Takeaways

• A system of linear inequalities is solved by finding the region where every inequality’s shaded area overlaps.
• Use dashed boundary lines for < and >, and solid lines for ≤ and ≥.
• The fastest shading test is plugging a point like (0, 0) into each inequality.
• Parallel boundary lines with opposite shading directions often mean the system has no solution.
• Always verify a solution point against every inequality, not just one.
• Graphing the boundaries first, then shading, then locating the overlap is a reliable, repeatable four-step method.