1Cademy - Example: Interpreting and Graphing $$x \leq 1$$

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Interval Notation for Inequalities

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Example: Interpreting and Graphing $x \leq 1$

When representing the inequality $x \leq 1$ , the solution set comprises all numbers that are less than or equal to $1$ . Since the number $1$ is a valid solution, a square bracket is positioned at $1$ on the number line to show it is included. All numbers to the left of $1$ are then shaded to represent that valid solutions continue endlessly downwards. In interval notation, this unbounded lower end is represented by the negative infinity symbol $-\infty$ , making the interval $(-\infty, 1]$ .

Updated 2026-06-03

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OpenStax Intermediate Algebra

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As a logistics coordinator, you are tracking a shipment where the allowed delay time ( $x$ , in hours) must be less than or equal to 1 hour, written mathematically as $x \leq 1$ . When documenting this rule in your tracking system using interval notation, which of the following correctly represents all valid solutions?
In a chemical processing plant, a safety valve is configured to trigger if the pressure fluctuation ( $x$ ) is less than or equal to 1 psi ( $x \leq 1$ ). To ensure accurate documentation in the plant's safety log, match each mathematical symbol or graphical action with the specific part of the safety rule it represents.
Graphing Inventory Thresholds
In a professional quality control environment, a technician monitors a machine's tolerance level ( $x$ ) defined by the inequality $x \leq 1$ . True or False: When representing this tolerance using interval notation, the correct form is $(-\infty, 1]$ , where the square bracket indicates that a measurement of exactly 1 is included in the set of acceptable values.
An operations manager is mapping out a timeline for a task that must be completed in 1 hour or less, represented by the inequality $x \leq 1$ . When graphing this acceptable timeframe on a number line, a square bracket is placed at 1, and all numbers to the ____ of 1 are shaded to show that the valid durations continue endlessly downwards.
As a quality assurance specialist, you are creating a visual guide to train new inspectors on logging part defect sizes. The tolerance for a specific part is less than or equal to 1 millimeter, written as $x \leq 1$ . Arrange the procedural steps in the correct order to accurately graph this inequality on a number line and document it in interval notation.
Visualizing Operational Safety Thresholds

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