What are the steps used to solve a linear inequality?

Solving a linear inequality is very similar to solving a linear equation, with one crucial difference: if you multiply or divide by a negative number, you must reverse the inequality sign. Here are the steps:

1. Simplify both sides of the inequality: This involves distributing any terms and combining like terms on each side of the inequality sign.

2. Rearrange the inequality: Move all terms containing the variable (usually 'x') to one side of the inequality and all constant terms to the other side. This is achieved using addition and subtraction. Remember that adding or subtracting the same value from both sides of an inequality does not change the inequality. For example, if you start with 2x + 3 > 7, you would subtract 3 from both sides to get 2x > 4.

3. Isolate the variable: Divide both sides of the inequality by the coefficient of the variable. Important: If you divide by a negative number, you must reverse the inequality sign. For instance, if you have -3x > 9, dividing both sides by -3 gives you x < -3 (notice the flipped inequality sign). However, if you have 3x > 9, dividing both sides by 3 gives you x > 3.

4. Write the solution: Express the solution in terms of the variable and the inequality symbol. This shows the range of values that satisfy the inequality. For example, x < -3 means that any number less than -3 will make the original inequality true.

Example:

Solve the inequality 5x - 2 ≤ 8x + 4

1. Simplify: The expressions on each side are already simplified.

2. Rearrange:

   • Subtract 5x from both sides: -2 ≤ 3x + 4
   • Subtract 4 from both sides: -6 ≤ 3x

3. Isolate the variable: Divide both sides by 3: -2 ≤ x

4. Write the solution: -2 ≤ x which is the same as x ≥ -2. This means all numbers greater than or equal to -2 will satisfy the inequality.