Solving Inequalities and Representing Solutions on a Number Line

Solve the following inequality and represent the solution set on a number line.

Steps to Solve an Inequality

1. Simplify both sides of the inequality by combining like terms.
2. Isolate the variable by performing the same operation on both sides of the inequality, keeping in mind that multiplying or dividing by a negative number reverses the inequality sign.
3. Represent the solution set on a number line.

Example:

Solve the inequality 2x + 3 < 7 and represent the solution set on a number line.

1. Simplify: 2x + 3 < 7
2. Isolate x:
   • Subtract 3 from both sides: 2x < 4
   • Divide both sides by 2: x < 2
3. Represent on a number line:
   • Draw a number line.
   • Mark the point 2 on the number line.
   • Since the inequality is 'less than', use an open circle at 2 (because 2 is not included in the solution set).
   • Shade the part of the number line to the left of 2, representing all values of x that are less than 2.

Key Points:

• The solution set of an inequality represents all the values of the variable that satisfy the inequality.
• The number line provides a visual representation of the solution set.
• The open circle on the number line indicates that the endpoint is not included in the solution set, while a closed circle indicates that the endpoint is included.
• When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.