How to Graph Solutions to Linear Inequalities?
Step 1: Identifying the inequality you want to graph
Step 2: Rewrite the inequality
Step 3: Draw a number line and label it
Step 4: Draw a circle on the number line
Step 5: Draw an arrow pointing in the direction
Step 6: Label the shaded region to indicate which inequality it represents
Step 7: Repeat these steps for any additional inequalities that you need to graph
Graphing Solutions to Linear Inequalities, Examples 1
Solution:
Graphing Solutions to Linear Inequalities, Examples 2
Solution:
To graph this inequality on a number line, we need to isolate the variable \(x\) on one side of the inequality. Adding \(6x\) to both sides, we get:
Tutor-style math help
Graph Solutions to Linear Inequalities: what to notice and how to work it
Inequalities skill
Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.
What to notice first
Graph the boundary first, then test a point to decide which side to shade.
Common student mistake
Do not use a solid boundary for a strict inequality. Dotted boundaries mean the boundary itself is not included.
Key formulas and cues
\(a<b\)
\(a\le b\)
\(\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}\)
\(|x-a|<b \Rightarrow a-b<x<a+b\)
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
Example: \(-3x>12\)
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Answer: \(x<-4\)
Keep the sign
Example: \(x+5\le9\)
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Answer: \(x\le4\)
Try one before moving on
Try: Solve \(-2x\le10\).
Answer: \(x\ge-5\). Divide by -2 and flip the sign.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Graph Solutions to Linear Inequalities: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Solve \(x-2>5\).
2. When must the inequality sign flip?
3. \(x\le3\) uses:
\(2x + 2 ≥ -1\)
Now, subtracting \(2\) from both sides, we get:
\(2x ≥ -3\)
Finally, dividing both sides by \(2\), we get:
\(x ≥ -\frac{3}{2}\)
To graph this solution on a number line, we start by drawing a line and marking a point at \(-\frac{3}{2}\). Since \(x\) is greater than or equal to \(-\frac{3}{2}\), we shade the region to the right of the point. The graph should look like this:
Exercises for Graphing Solutions to Linear Inequalities
Graph the solution of each inequality.
- \(\color{blue}{4x+3\ge 2x+11}\)
- \(\color{blue}{\frac{x}{9}-6\le 4}\)
Original price was: $109.99.$54.99Current price is: $54.99.
- \(\color{blue}{4x+3\ge 2x+11}\)
- \(\color{blue}{\frac{x}{9}-6\le 4}\)
Original price was: $109.99.$54.99Current price is: $54.99.
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