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Linear Inequality

Linear Inequality

An inequality that can be written in the form "linear polynomial > linear polynomial" or "linear polynomial > constant". The > sign may be replaced by <, ≤, or ≥.

The following are examples of linear inequalities: 2x – 3 < 5, 4a + 9 ≥ 8 – 9a, and 2x + 5y ≤ 1.

 

 

See also

Linear polynomial, constant

Key Formula

ax+b<cax + b < c
Where:
  • aa = Coefficient of the variable (a ≠ 0)
  • xx = The variable being solved for
  • bb = A constant term
  • cc = A constant on the other side of the inequality
  • << = Can be replaced by >, ≤, or ≥

Worked Example

Problem: Solve the linear inequality 3x − 7 ≤ 11 and express the solution.
Step 1: Add 7 to both sides to isolate the term with the variable.
3x−7+7≤11+7  ⟹  3x≤183x - 7 + 7 \leq 11 + 7 \implies 3x \leq 18
Step 2: Divide both sides by 3. Since 3 is positive, the inequality sign stays the same.
3x3≤183  ⟹  x≤6\frac{3x}{3} \leq \frac{18}{3} \implies x \leq 6
Step 3: Write the solution in interval notation or on a number line. The solution includes 6 and all numbers less than 6.
x∈(−∞,6]x \in (-\infty, 6]
Answer: x ≤ 6, meaning every value from negative infinity up to and including 6 satisfies the inequality.

Another Example

Problem: Solve −2x + 4 > 10.
Step 1: Subtract 4 from both sides.
−2x+4−4>10−4  ⟹  −2x>6-2x + 4 - 4 > 10 - 4 \implies -2x > 6
Step 2: Divide both sides by −2. Because you are dividing by a negative number, you must flip the inequality sign.
−2x−2<6−2  ⟹  x<−3\frac{-2x}{-2} < \frac{6}{-2} \implies x < -3
Step 3: Write the solution. Note the open interval — −3 itself is not included.
x∈(−∞,−3)x \in (-\infty, -3)
Answer: x < −3. Any value strictly less than −3 makes the original inequality true.

Frequently Asked Questions

When do you flip the inequality sign?
You flip (reverse) the inequality sign whenever you multiply or divide both sides of the inequality by a negative number. For example, dividing both sides of −4x > 12 by −4 gives x < −3. If you multiply or divide by a positive number, the sign stays the same.
How do you graph a linear inequality on a number line?
First solve for the variable. Then mark the boundary value on the number line. Use an open circle (○) if the inequality is strict (< or >), or a closed circle (●) if it includes equality (≤ or ≥). Finally, shade the region in the direction of all values that satisfy the inequality.

Linear Equation vs. Linear Inequality

A linear equation uses an equals sign and typically has exactly one solution (e.g., 2x + 1 = 7 gives x = 3). A linear inequality uses <, >, ≤, or ≥ and has infinitely many solutions forming a range (e.g., 2x + 1 < 7 gives x < 3). Solving both follows similar steps, but with inequalities you must flip the sign when multiplying or dividing by a negative number.

Why It Matters

Linear inequalities model real-world constraints where exact equality isn't required — budgets, speed limits, minimum requirements, and capacity limits are all naturally expressed as inequalities. They are foundational in algebra and extend into systems of linear inequalities, which form the basis of linear programming used in business and engineering optimization. Mastering them prepares you for graphing inequalities in two variables, where solutions become shaded regions on the coordinate plane.

Common Mistakes

Mistake: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
Correction: Every time you multiply or divide both sides by a negative value, reverse the direction of the inequality. A good habit is to circle or highlight the step where this happens so you don't miss it.
Mistake: Using a closed circle (●) on the number line for strict inequalities (< or >), or an open circle (○) for ≤ or ≥.
Correction: Closed circles mean the endpoint is included (≤ or ≥). Open circles mean it is not included (< or >). Match the circle type to the symbol in your solved inequality.

Related Terms