7. Solving Linear Inequalities and Compound Inequalities

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1 7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing a constant on the inequalities and expressing the solution set. However, if you want to practice with solving linear equations, you can refer to the previous topic. (Topic 6) This handout will show some examples on how to solve linear inequalities and compound inequalities and how to express the solution sets of inequalities. Solve Linear Inequalities Example (): x + 8 > 6 Solution: x > 6 8 x > x > x > Subtract 8 on each side Divide on each side. Do not reverse the inequality symbol. x x > Example (): x 5x + Solution: x + 5x + + x 5x + 5 x 5x 5x 5x + 5 x 5 x 5 5 x 5 x x Add on each side Subtract 5x on each side Divide - on each side; reverse the inequality symbol (when divide or multiply a negative number). This instructional aid was prepared by the Learning Commons at Tallahassee Community College

2 Example (): 6 ( + x ) < Solution: 8 + x < x + 6 < x < 6 x < x < x < 6 Remove the parenthesis by multiplying 6 to and x. Subtract 6 on each side Divide on each side. Do not reverse the inequality symbol. x x < 6 Example (): ( w ) ( w) Solution: ( ) ( w ) ( )( w) ( ) ( w ) ( w) w + w w 7 w w 9 w { w w } Multiply on each term to simplify the inequality Remove parenthesis. Multiply - to ( w) Add 7 on each side Divide on each side. Do not reverse the inequality symbol. This instructional aid was prepared by the Learning Commons at Tallahassee Community College

3 Example (5): 5z + 5z > 5 5z + 5z 5 Solution: ( 5) > ( 5) ( 5z ) > 5( 5z) + 5 z > + 5z 5 z + > + + 5z 5 z > + 5z 5z 5z > + 5z 5z z > z < z < 5 Find LCD=5. Multiply 5 to each term Distribute property to remove the parenthesis Add on each side Subtract 5z on each side Divide - on each side. Reverse the inequality symbol. z z < 5 Interval Notation - [,) Use the open parentheses ( ) if the value is not included in the graph, i.e. greater than (>) or less than (<). Use the brackets [ ] if the value is part of the graph, i.e. greater than or equal to ( ). Whenever there is a break in the graph, write the interval up to the point. Then write another interval for the section of the graph after that part. Put a union sign between each interval to "join" them together. This instructional aid was prepared by the Learning Commons at Tallahassee Community College

4 Solve Compound Inequalities (two inequalities joined by and or or ) Example (): x < and x Solution: When solving compound inequalities, we usually graph them on the number lines to get the solution set. x < x x < and x - - Interval Notation: [,) * When two inequalities joined by and, that means interception of the solutions. * Look for overlapping of the graph. from the overlapping segment, if any. Example (): x < or x Solution: When solving compound inequalities, we usually graph them on the number lines to get the solution set. x < When two inequalities joined by or, that means union of the solutions. x - x < or x - Interval Notation: (, ) * Look for everything shaded on the graph. from the number line. This instructional aid was prepared by the Learning Commons at Tallahassee Community College

5 Example (): x + < 9 and x > 7 Solution: We need to solve each inequality before we can place them on the number lines. x + < 9 x < 8 8 * When two inequalities joined by and, that means interception of the solutions. x + > 7 + x > 8 x > 8 * Look for overlapping of the graph. x < 8and x > Interval Notation: (,8) 8 from the overlapping segment, if any. Example (): x + < or x > 8 Solution: We need to solve each inequality before we can place them on the number lines. x + < x < 6 * When two inequalities joined by or, that means union of the solutions. x > x < or x > Interval Notation:,) ( (, ) * Look for everything shaded on the graph. from This instructional aid was prepared by the Learning Commons at Tallahassee Community College 5

6 Example (5): 5 < x + < 9 Solution: This is a three-part inequality. We will solve this inequality a little different than previous examples. However, our goal is to isolate the variable x in the middle. 5 < x + < 9 8 < x < 6-8 ( 8,6) 6 *To isolate the variable x, we need to subtract in the middle as well as two sides. *State the solution in interval notation. (you can graph the solution on the number line to help you write out the interval notation.) Example (6): < 7 x 9 Solution: This is a three-part inequality, so our goal is to isolate the variable x in the middle. 7 < 7 7 x < x 9 x > > x *The first thing we need to do to isolate the variable x is subtracting 7 in the middle as well as two sides. *Next we need to divide - in the middle as well as two sides and Reverse the inequality symbol. * State the solution in interval notation. (you can graph the solution to help you write out the interval notation.) [,) This instructional aid was prepared by the Learning Commons at Tallahassee Community College 6

7 Exercises: Solve the following inequalities. Write the solution in interval notation.. x + or x +. < 5 x. t 5t ( t + ). x x + < 5 5. ( x) x Answers:. (, ] [, ). [,). (, ]. (,) 5, 5. This instructional aid was prepared by the Learning Commons at Tallahassee Community College 7