The document discusses solving linear equalities and inequalities with one variable. It defines key terms like equations, inequalities, and linear equations. It then provides steps for solving different types of linear equations and inequalities by collecting like terms, adding/subtracting the variable term to one side, and multiplying/dividing both sides by constants. The document also explains how to graph solutions to inequalities on a number line, indicating open and closed circles based on the inequality symbols. Examples are provided of solving and graphing various linear equalities and inequalities with one variable.

2. DEFINITION: Equation Inequality Solution of an Equation Two expressions set equal to each other Linear Equation An equation that can be written in the form ax + b = 0 where a and b are constants A value, such that, when you replace the variable with it, it makes the equation true. (the left side comes out equal to the right side) Mathematical expressions that use the symbols ( <, ≠, >, ≥, etc.)

3. Solve for x . Solving a Linear Equation in General Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations. *Inverse of sub. 10 is add. 10

4. Solve for x . *Inverse of add. 7 is sub. 7

5. *Inverse of div. by 3 is mult. by 3 Solve for x

6. Inverse of mult. by 5 is div. by 5 Solve for x .

7. Solve. 4 x + 6 = x Solving Equations with Variables on Both Sides 4 x + 6 = x – 4 x – 4 x 6 = –3 x To collect the variable terms on one side, subtract 4x from both sides. Since x is multiplied by -3, divide both sides by – 3. – 2 = x You can always check your solution by substituting the value back into the original equation. 6 – 3 – 3 x – 3 =

8. Solve. 3 b – 2 = 2 b + 12 3 b – 2 = 2 b + 12 – 2 b – 2 b b – 2 = 12 + 2 + 2 b = 14 Since 2 is subtracted from b, add 2 to both sides. To collect the variable terms on one side, subtract 2b from both sides.

9. Solve. 3 w + 1 = 3 w + 8 1 ≠ 8 No solution. There is no number that can be substituted for the variable w to make the equation true. 3 w + 1 = 3 w + 8 – 3 w – 3 w To collect the variable terms on one side, subtract 3w from both sides.

11. SOLVING LINEAR INEQUALITY IN ONE VARIABLE

12. An inequality compares two expressions using <, >,  , or  . is less than is greater than is greater than or equal to is less than or equal to Fewer than, below More than, above At most, no more than At least, no less than An inequality that contains a variable is an algebraic inequality . Symbol Meaning Word Phrases < > ≤ ≥

13. A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set .

14. Solve the inequality. Solving Inequalities by Adding or Subtracting x + 3 > –5 x + 3 > –5 – 3 –3 x > –8 Since 3 is added x, subtract 3 from both sides.

15. m – 4 ≥ –2 m – 4 ≥ –2 + 4 +4 m ≥ 2 Since 4 is subtracted from m, add 4 to both sides. Solve Solving Inequalities by Adding or Subtracting

16. When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true.

17. b ≥ –5 – 9 b ≤ 45 Divide both sides by –9; ≤ changes to ≥. Solve Solving Inequalities by Multiplying or Dividing ≥ 45 – 9 –  9 b – 9

18. 48 < a , or a > 48 12 < Multiply both sides by 4. Solve Solving Inequalities by Multiplying or Dividing a 4 4 • 12 < 4 • a 4

19. 80 > b , or b < 80 16 > Multiply both sides by 5. Solve b 5 5 • 16 > 5 • b 5

20. Solve the inequality. x + 4 > –2 x + 4 > –2 – 4 –4 x > –6 Since 4 is added x, subtract 4 from both sides.

21. GRAPHING INEQUALITY IN ONE VARIABLE

22. You can graph the solution set on a number line. The symbols < and > indicate an open circle while ≥ and ≤ indicate a close circle.

23. This open circle shows that 5 is not a solution. a > 5 The symbols ≤ and ≥ indicate a closed circle. This closed circle shows that 3 is a solution. b ≤ 3

24. Graph each inequality. – 3 –2 –1 0 1 2 3 A. –1 > y Draw an open circle at –1. The solutions are all values of y less than –1, so shade the line to the left of –1. B. z ≥ –2 – 3 –2 –1 0 1 2 3 Graphing Inequalities 1 2 Draw a closed circle at –2 and all values of z greater than  2 . So shade to the right of –2 . 1 2 1 2 1 2

25. Graph each inequality. – 3 –2 –1 0 1 2 3 A. n < 3 B. a ≥ –4 – 6 –4 –2 0 2 4 6 Example 3 Draw an open circle at 3. The solutions are all values of n less than 3, so shade the line to the left of 3. Draw a closed circle at –4. The solutions are all values greater than –4, so shade to the right of –4.

26. Solve and graph. 1. –14 x > 28 2. < 15  5 3. 18 < –6 x x < –2 q ≥ 40 – 3 > x x < 45 4. – 2 0 2 50 40 45  40 45 x 3 q 8

27. 6. –3 < y Graph each inequality. 5. m ≤ 1 0 º 1 2 3 – 1 – 2 3 – – – – 0 1 2 3 1 2 3