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Closed or Open

The solutions of $x \ge 1$ include 1 itself, because 1 is greater than or equal to 1. On the number line, the dot at 1 is closed (filled in) to show that 1 is a solution. The shading runs to the right.

$x ≥ 1: a filled dot at 1, shaded to the right$

x ≥ 1: a filled dot at 1, shaded to the right

The dot shows the difference between < and ≤. It is open for a strict sign (< or >) and closed for a sign with "or equal to" (≤ or ≥). For example, $x \le -2$ is a filled dot at -2 with shading to the left.

$x ≤ -2: a filled dot at -2, shaded to the left$

x ≤ -2: a filled dot at -2, shaded to the left

Challenge

Easy

Show the solutions of $x \le 4$ on the number line.

Try it yourself

x \le 4

Show the solutions

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Inequalities & Absolute Value

1What an Inequality Is

The Four Signs A Picture on a Line Reading It Backwards Recap: What an Inequality Is

4Two-Step Inequalities

Two Steps Subtract, Then Flip With a Fraction A Bracket on the Way Recap: Two-Step Inequalities

7Between Two Numbers

Two Conditions at Once Solving a Chain Two Moves on All Parts When the Chain Flips Chains with Fractions Recap: Between Two Numbers

10Absolute Value Inequalities

Less Than: Between Between, Shifted Greater Than: Outside Outside, Shifted Isolate, Then Split Always or Never Recap: Absolute Inequalities

2Adding and Subtracting

Add to Both Sides Solve, Then Draw Recap: Adding and Subtracting

5Variables on Both Sides

x on Both Sides Gather Toward the Larger Brackets First Always True, Never True Recap: Variables on Both Sides

8One or the Other

Either Works Solving Each Part And versus Or Recap: One or the Other

11From Words to Inequalities

At Least, At Most More Than A Budget That Shrinks Two Limits Recap: From Words