Solve linear inequalities on a number line

• Apply the same operation to both sides while tracking order direction.
• Explain the sign reversal caused by multiplication or division by a negative number.
• Represent a solution set with endpoint inclusion and a number-line ray.

Build the idea

Worked example

1. Step 1

   Subtract four from both sides. Translation by the same amount preserves the order and leaves negative three x less than or equal to nine.

   $$-3x\le9$$

2. Step 2

   Divide by negative three and reverse the inequality because the scaling reflects the number line. The boundary becomes negative three.

   $$x\ge-3$$

3. Step 3

   Use a closed point at negative three because equality is allowed, then shade to the right for values greater than the boundary.

   $$[-3,\infty)$$

• Keeping the less-than-or-equal symbol unchanged after division by negative three.

  Negative scaling reflects order. Reverse the symbol at the exact step where multiplication or division by a negative occurs.

• Drawing an open circle even though the original inequality includes equality.

  The bar under the symbol includes the boundary value, so the endpoint must be closed and the interval must use a square bracket.

Solve five minus two x is greater than one, graph the answer, and test an integer from the solution set.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   An inequality describes a set of values rather than one balance point. Equality-preserving operations still work, except multiplying or dividing by a negative reverses the order. A statement such as x greater than three names every number to the right of three. Solving an inequality transforms the statement while preserving exactly that set of valid inputs. Apply the same operation to both sides while tracking order direction.

2. Part 2

   Visual model and equations

   Adding or subtracting the same amount shifts both quantities without changing which is larger. Multiplication by a positive amount scales the line in the same direction, but multiplication by a negative reflects the number line through zero. Place negative two to the left of one. Multiplying both values by negative one sends two to the right of negative one, so the former smaller value becomes larger and the order symbol must reverse. Negative two is less than one, so after multiplying by negative one, two is greater than negative one. x is greater than or equal to negative three.

   • Negative two is less than one, so after multiplying by negative one, two is greater than negative one.
   • x is greater than or equal to negative three.
3. Part 3

   Worked example

   Solve negative three x plus four is less than or equal to thirteen, graph the result, and test one included value. Remove the added four, divide by negative three while reversing the symbol, and interpret the inclusive endpoint on a number line. Subtract four from both sides. Translation by the same amount preserves the order and leaves negative three x less than or equal to nine. Divide by negative three and reverse the inequality because the scaling reflects the number line. The boundary becomes negative three. Use a closed point at negative three because equality is allowed, then shade to the right for values greater than the boundary. The solution is every x greater than or equal to negative three. Choose x equals zero from the shaded region. The original left side becomes four, which is less than or equal to thirteen, so the tested value supports the solution direction.

   • Negative three x is less than or equal to nine.
   • x is greater than or equal to negative three.
   • The interval starts at included negative three and continues to positive infinity.
4. Part 4

   Checks, practice, and scope

   That reflection swaps left and right, so the inequality symbol reverses. A test value from the proposed solution region then checks whether the direction and endpoint were handled correctly. Keeping the less-than-or-equal symbol unchanged after division by negative three. Negative scaling reflects order. Reverse the symbol at the exact step where multiplication or division by a negative occurs. Drawing an open circle even though the original inequality includes equality. The bar under the symbol includes the boundary value, so the endpoint must be closed and the interval must use a square bracket. Solve five minus two x is greater than one, graph the answer, and test an integer from the solution set. Subtract five first, then divide by negative two and reverse the strict inequality. The solution is x less than two, shown with an open point at two and shading to the left. Inequality solving preserves a whole set of values, not just one number. Only multiplication or division by a negative reverses the order symbol. Endpoint style and shading direction encode inclusion and order on the graph. This lesson covers one linear inequality, not compound, absolute-value, polynomial, or rational inequalities. A single test value supports the final region but does not replace the algebraic equivalence of every solving step.

Scope and limitations

• This lesson covers one linear inequality, not compound, absolute-value, polynomial, or rational inequalities.
• A single test value supports the final region but does not replace the algebraic equivalence of every solving step.