Generate values that satisfy an inequality

An inequality with a variable describes a set of possible values rather than a single answer; a value satisfies it when substitution makes the comparison true. Learners generate and verify whole-number, decimal, or fraction values that make an inequality true, represent the solution region on a number line, and determine whether an endpoint is included for $<, >, \leq,$ or $\geq$ ; complex inequality solving and interval notation are beyond this scope.

Detailed Explanation: Generate values that satisfy an inequality

An inequality can have many correct answers. A number satisfies an inequality when replacing the variable with that number makes the statement true.

Example

Find values of $x$ that satisfy

x+3\le 8

Step 1: Find the boundary value.

Subtract $3$ from both sides:

x\le 5

This tells us that $x$ can be $5$ or any number less than $5$ .

Step 2: Generate possible values.

Some values that satisfy the inequality are

x=5,\qquad x=4,\qquad x=2.5,\qquad x=\frac12.

Step 3: Verify a value by substitution.

Try $(x=2.5)$ :

2.5+3=5.5

Since $(5.5\le 8)$ is true, $(x=2.5)$ satisfies the inequality.

Try $(x=6)$ :

6+3=9

Since $(9\le 8)$ is false, $(x=6)$ does not satisfy the inequality.

Step 4: Show the solution on a number line.

Because $x$ can equal $5$ , use a filled dot at $5$ . Shade to the left because numbers less than $5$ also work.

<==========●---------------->
           5

So the solutions are all numbers less than or equal to $5$ . The endpoint $5$ is included because the inequality uses $\le$ .

Learn by doing: Generate values that satisfy an inequality

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Function Domain/Range Definition - Inequality to Number Line (With Union)

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