Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with Fractions Section. Linear Inequalities in One-Variable Answers
Section. Additive Property of Equality Eamples: Solve each equation. a) 8 + = 7 8 + = 7 + 8 + 8 = 39 Add 8 to both sides of the equation to undo subtracting 8 from. b) 0 34 = + 7 0 34 = + 7 4 = + 7 7 7 = First combine like-terms on the left side, by subtracting 34 from 0. Subtract 7 from both sides of the equation to undo adding 7 to. Homework. In your own words, describe a linear equation. How does one recognize that an equation is linear? Give two eamples of linear equations.. What is the inverse operation of addition? 3. What is the inverse operation of subtraction? Solve the equation or simplify the epression. 4. a) + 3 = 8 b) y 4 = 67 c) + 8 d) 9 = k 6 e) 6 7 + + Solve each equation. Show a check for every third problem.. y + 7 = 6. p 3 = 6 7. p + 4 = 7 8. 8 = z + 4 9. 6 = m 0. 3 = 48 + n. 7 + = 7. z 36 = 00 3. + w = 4. 0 = + 8..4 = m.9 6. 7 + = 9 Section. Additive Property of Equality
7. 6 = + c 8. 3 + p = 3 9. y + 6 = 9 0. 8 = +. 3 =. m + 34 = 0 3. 9 = + Section. Additive Property of Equality
Section. Multiplicative Property of Equality Eamples: Solve each equation. a) 0 = 0 = 0 0 Divide both sides of the equation by 0 to undo multiplying by 0. = 4 Simplify the fraction to. 0 4 b) = 8 = 8 8( ) = 8 8 8( ) = 8 8 6 = The negative sign in front of the fraction can be applied to either the numerator or the denominator. But not both at the same time. In this problem, it s easiest to apply it to the denominator, since that is a constant factor. Multiply both sides of the equation by 8 to undo dividing by 8. Homework. What is the inverse operation of multiplication?. What is the inverse operation of division? Solve the equation or simplify the epression. 3. a) 3 = 8 b) + 8 + c) p = 0 d) 4y + 3y e) 3 = 4 Solve each equation. Show a check for every third problem. Be sure to notice the operation in the equation before you solve. 4. 4 m =. 0 p = 40 r 6. = 0 3 7. = 8. y = 7 9. 4 = 6w 0. 9 =. 8 = p 3 Section. Multiplicative Property of Equality 3
. a = 48 3. 6 = 9z 4..3y = 6. = 6 6. w 7 = 7. 7 = 4m 8. 6c = 0 9. = 7 0. 0 = 6w. 4 = 7 a. 4 = 3. 6 = 4 p 4. 8n = 3. = 0 + y 6. 9 = 8z 7. = 0 k 8. = 0 Section. Multiplicative Property of Equality 4
Section.3 Linear Equations in One-Variable.3 Linear Equations in One-Variable Worksheet Eamples: Solve each equation. a) 7 = 3( 6) 7 = 3( 6) 8 = 3 + 8 8 = 0 3 8 = 0 3 + 3 + 3 = 0 Distribute 3. Combine like-terms. Add 3 to both sides of the equation to get the variable terms on one side of the equation and the constant terms on the other side of the equation. 0 = Divide both sides by to undo multiplying by. = 4 b) 7 = ( 3 + ) ( ) 7 = 3 + 7 = 6 6 = 6 6 = 6 6 6 = Distribute. Combine like-terms. This equation is an identity, since both sides are eactly the same. To satisfy the equation, can be any real number. If you notice this at this point, you may stop solving and write the answer. Subtract 6 from both sides of the equation to get the variable terms on one side of the equation and the constant terms on the other side of the equation. Since there are no variables left and the statement is true, the answer is all real numbers. c) 0 + 4 = ( + ) ( ) 0 + 4 = + 0 + 4 = 0 + = 0 0 + = 0 0 0 = Distribute. Combine like-terms. This equation is a contradiction, since both sides have the same variable term, but different constants. There is no number that will satisfy the equation. If you notice this at this point, you may stop solving and write the answer. Subtract 0 from both sides of the equation to get the variable terms on one side of the equation and the constant terms on the other side of the equation. Since there are no variables left and the statement is false, the answer is no solution. Section.3 Linear Equations in One-Variable
Homework. In your own words, describe how solve a linear equation.. In your own words, describe how to determine if a linear equation has no solution. 3. In your own words, describe how to determine if all real numbers are solutions to a linear equation. Solve the equation or simplify the epression. 4. a) 3 + = 40 b) + 3 + 40 c) y + y 7 d) 4 = Solve each equation. Show a check for every third problem. Give simplified answers.. = 6. 6 + 8 = 3y 7. 4 p + 6 = 3 8. 9 = 4w 6 9. = 0 + 0. 4 3+ = 6. ( ) = 4. = 4( ) 3. ( ) 3 + + = 7 4. = 3( + ). a ( a ) 7 3 + 4 = 3 + 3 6. ( ) + + 4 = 6 7. 8 = 6 ( + 4) 8. 3 = 7 + ( ) 9. ( ) 3 + 6 + 8 = 4 0. + 3 =. 9 = 6. 8y + = 3y + 4 3. 4 = 4 4. 0 = + 0. + 8 = ( + 4) 6. 4 + = 7 7. 3 = + 3 8. 7 p = 4( p 3) Section.3 Linear Equations in One-Variable 6
4 + = 3 9. ( ) 30. 8 = 4( + 8) 3. 9( + ) = 0 4( + 7) 3. 6( m ) = 3 + ( m 9) 33. 4 0 = 8 3( + ) 7 4 6 3 = 7 + 6 34. ( ) 3. + 8 = 9( + 3) 36. 7 + ( 3) = 4 ( 6 ) 37. 6 = 4( n 7) 38. ( ) 4 + = 3 39. 7 + = 9 + 6( + ) 40. 9 ( z 6) z = 7 + 3( z 8) 4. ( ) + = 3 4. 8( 3) + 3 = + 7( + ) 43. + = 6( ) Section.3 Linear Equations in One-Variable 7
Section.4 Linear Equations in One-Variable with Fractions.4 Equations and Epressions Worksheet Eample: Solve each equation. 4 9 a) = 3 0 4 9 3 8 = 4 4 9 3 3 8 = Distribute 4 3. 3 4 4 9 = 3 3 8 3 = 3 To multiply both terms on the left side of the equation by 4 3, cancel factors common to numerator and denominator in each multiplication. 6 6 3 6 6 = 3 3 6 6 3 6 6 = 3 4 9 = 6 4 9 = 6 4 4 9 = 9 = + 30 + 30 = = = The LCD of simplified equation is 6. Since this is an equation, you may multiply each side of the equation by the LCD to get rid of the fractions. Multiply all terms on both sides of the equation by 6 or 6. Again, cancel factors common to numerator and denominator in each multiplication. It may be helpful to write integers with a denominator of (as with and in this problem). Subtract 4 from both sides of the equation to get the variable terms on one side of the equation. Add 30 to both sides of the equation to get the constant terms on one side of the equation, as well as to undo subtracting 30. Divide both sides of the equation by, to undo multiplying by. Homework. In your own words, describe how to solve a linear equation that contains fractions. First, state the number of terms in each equation as it is written. Second, if the equation has a factor to distribute, distribute and then state how many terms are in the equation. 3 6. a) = w 4 Section.4 Linear Equations in One-Variable with Fractions 8
9 3 b) 4 ( ) + + = c) 3 = ( + 4) Solve each equation. Show a check for every third problem. Give simplified answers. 3. = 6 4 4.. 6. 3 7 = y 0 3 3p + = 7 4 3 6 4 = w ` 8 = 7 7. ( ) 8 3 8. = ( + ) + + = 9 3 9. 4 ( ) 0. 3 6 = 4 8 + = 3 4. ( ). 3. 4.. 6. 7. 6 + = 7 4 m 3 = m + 8 6 = 3 6 3 + = 3 4 3 3 + = 4 4 4 z + 3 = z 4 0 6 8. ( 0 ) = ( 3) 4 3 4 7 n + = n + 9. ( ) 3 9 0. 4 ( 6 + 4) = ( 9 ) Section.4 Linear Equations in One-Variable with Fractions 9
Section. Linear Inequalities in One-Variable Eample: Solve each equation. 0 8 3 0 a) ( ) ( ) 0 8 3 0 + 4 3 0 6 3 0 6 3 0 + 3 + 3 6 0 6 0 6 6 36 Distribute. Combine like terms. Add 3 to both sides of the equation to get the variable terms on one side of the equation. Subtract 6 from both sides of the equation to get the constant terms on one side of the equation, as well as to undo adding 6. 36 8 Divide both sides of the equation by to undo multiplying by AND switch the direction of the inequality sign. You must switch the inequality sign whenever you divide (or multiply) by a negative value in order to solve an inequality. The solution is all real numbers that are greater or equal to 8. 0 8 Homework Define each word below in your own words and then give a few eamples.. a) Epression b) Equation c) Inequality. Describe in your own words how to solve an inequality. 3. Describe in your own words how to determine when you switch the direction of an inequality sign when solving an inequality. 4. Describe in your own words when you would use a closed (solid) circle/dot when graphing an inequality.. Describe in your own words when you would use an open (hollow) circle/dot when graphing an inequality. 6. Describe in your own words how to determine which interval an inequality represents, in other words, which way the line goes, when graphing an inequality. Graph each inequality on a number line. 7. > 4 8. < 9. 0 0. 8.. < 3. 3 > 4. > 0. 0 Section. Linear Inequalities in One-Variable 0
Solve each inequality. Graph the solution on a number line for every fifth problem. Check every fifth problem. 6. 3 4 > 8 7. + 8. 4 < 6 9. 3p + 9 0. 6 + 7 3. + 4. 4 7 a 3. < 7 3 4. ( y 6) 4. ( ) 3 7 > 9 6. 6 4( + ) 7 7. ( w + ) 6 4 8 8. 4 > ( ) 9 3 3 9. ( ) 3 + 0 3 0 30. ( z ) 3 + 0 9z 3. 7 < 3 + 3. 4 < + 33. 4 3 + 7 3 7 3 34. + > 8 6 4 3. ( q ) 7 4 q 8 36. ( ) 4 + 7 37. m 8 m 8 38. 4 3 < 4 3 39. 0 0 + 40. 3 0 < 3 8 4. n + 4 < n 4. > 9 3 9 3 3 + 4 9 43. ( ) ( ) 44. ( y + 4 ) y ( y + ) 4. ( ) 0 3 3 0 7 9 3 + 4 46. ( c + 4 ) c ( c ) 8 3 3 3 4 3 Section. Linear Inequalities in One-Variable
47. ( + 4 ) < ( ) 3 3 4 3 48. ( ) 4 + 7 3 > 0 49. ( 8) 3 7 + 4 0 0. ( w ) 44 6 w +. 4 6 ( + 3) ( + ) 3 3 Section. Linear Inequalities in One-Variable