Skill: 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 $\le$.