Systems of Inequalities

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Idea

A system joined by “and” asks for values that satisfy every condition at once. Solve each inequality separately and intersect the solution sets. An “or” instead calls for their union.

Visual guide

VISUAL GUIDEA solution must satisfy both inequalities

y = x
y = 2 − x

The shaded wedge is the intersection y ≥ x and y ≤ 2 − x. Both boundary lines are included. The point (0, 1) satisfies both; the region continues left beyond the plotting window.

Method

x\in A\text{ and }x\in B\quad\Longleftrightarrow\quad x\in A\cap B.

Draw intervals on the same number line and retain only the overlap. The endpoint is included only if all required conditions include it.

Worked example

Solve $x^2\le9$ and $x>1$ . The first gives $[-3,3]$ , the second $(1,\infty)$ . Their intersection is $(1,3]$ . For $\sqrt{x+3}/(x-2)$ , combine $x\ge-3$ with $x\ne2$ : the domain is $[-3,2)\cup(2,\infty)$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What satisfies both x > −1 and x ≤ 3?

x ≤ 3 or x > −1

−1 ≤ x < 3

−1 < x ≤ 3

Hint 1 · Find a starting point

“Both” means the overlap of the solution sets.

Hint 2 · Take the next step

Keep each endpoint’s original strict or inclusive condition.

Show the reasoning

Answer: −1 < x ≤ 3

The common region is (−1,3], excluding −1 and including 3.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

Adding solution intervals together by union solves an “or” question, not an “and” question. Some intersections are empty.

Check your understanding

Solve $2x-1\ge3$ and $x+4<7$ .

Show answer

$x\ge2$ and $x<3$ , so $[2,3)$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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