Inequality

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Inequality

  1. 1. In aueqlI yt
  2. 2. a cbax + 4 = 10 x + 4 < 10 x + 4 > 101. Less than is written as “ < “2. Greater than is written as “ > “3. Less than or equal to is written as “≤ “4. Greater than or equal to is written as “ ≥ “The signs of inequalityLinear Inequality with one variable
  3. 3. What is Linear Inequality with OneVariable ?Linear inequality with one variable isan open sentence with one variableand the first power which isconnected by sign of inequalityLinear inequality with one variable isan open sentence with one variableand the first power which isconnected by sign of inequality
  4. 4. How to solve these inequalities?x + 4 < 10 x + 4 > 10
  5. 5. Solving A Linear Inequality withOne VariableSketching the graph of solution in a number lineLook at the following number line and then answer the question below!What numbers are solutions of the inequality x < 3?Is 4 a solution of that inequality?Is 3 a solution of that inequality?Is 2 a solution of that inequality?Is 1 a solution of that inequality?Is 0 a solution of that inequality?Is -1 a solution of that inequality?Is -2 a solution of that inequality?Is -3 a solution of that inequality?Can youmention allsolutions of thatinequality?
  6. 6. Meanwhile, the solutions can be described onthe following number line.The graph of solution of x ≤ 3 isx = 3 on the line is not dottedbecause 3 is not a solutionx = 3 on the graph is dotted because3 is also a solution
  7. 7. Working out an Inequality by Addition or SubtractionLook at statement -4 < 1. This statement is true. The number line belowshows what happens if 2 is added to both sides.If 2 is added to both sides, then we obtain a statement -2 < 3. Thatstatement is also true.In the example above, adding 2 to both sides does not change the truthvalue of the statement.+2 +2
  8. 8. Now, Look at statement -3 < 1. This statement is true. The number linebelow shows what happens if 2 is substracted from both sides.If 2 is subtracted from both sides, then we obtain a statement -5 < -1.That statement is still true.-2 -2In the example above, subtracting 2 from both sides does not change thetruth of the statement.
  9. 9. Properties of addition orsubtracting in an inequality• If a certain number is added to or subtractedfrom both sides of an inequality, the symbol ofthe inequality does not change, and thesolution does not change, either.• The new linear inequality that we get if acertain number is added to or subtracted fromboth sides is called a linear inequalityequivalent to the original one.
  10. 10. 4 > -1... = 4 × 2 ... -1 × 2 = ... (both sides are multiplied by 2)... = 4 × 1 ... -1 × 1 = ... (both sides are multiplied by 1)... = 4 × 0 ... -1 × 0 = ... (both sides are multiplied by 0)... = 4 × -1 ... -1 × -1 = ... (both sides are multiplied by -1)-8 = 4 × -2 ... -1 × -2 = ... (both sides are multiplied by -2)-4 < -1... = -4 × 2 ... -1 × 2 = ... (both sides are multiplied by 2)... = -4 × 1 ... -1 × 1 = ... (both sides are multiplied by 1)... = -4 × 0 ... -1 × 0 = ... (both sides are multiplied by 0)... = -4 × -1 ... -1 × -1 = ... (both sides are multiplied by -1)8 = -4 × -2 ... -1 × -2 = ... (both sides are multiplied by -2)
  11. 11. Working out Inequality byMultiplication or DivisionConsider the statement 4 > 1, 4 > -1, -4 <-1, 8 < 12, -8 < 12, andthe statement -8 > -12. Those two statements are true. Fill in theblanks below. First fill it with a suitable number, and then fill itwith the sign “<“, “>”, or “=“.4 > 18 = 4 × 2 ... 1 × 2 = ... (both sides are multiplied by 2)... = 4 × 1 ... 1 × 1 = ... (both sides are multiplied by 1)... = 4 × 0 ... 1 × 0 = ... (both sides are multiplied by 0)... = 4 × -1 ... 1 × -1 = ... (both sides are multiplied by -1)-8 = 4 × -2 ... 1 × -2 = ... (both sides are multiplied by -2)
  12. 12. 8 < 12... = 8 : 4 ... 12 : 4 = ... (both sides are divided by 4)4 = 8 : 2 ... 12 : 2 = ... (both sides are divided by 2)... = 8 : ... 12 : = ... (both sides are divided by )-8 = 8 : -1 ... 12 : -1 = -12 (both sides are divided by -1)... = 8 : -2 ... 12 : -2 = ... (both sides are divided by -2)-8 < 12... = -8 : 4 ... 12 : 4 = ... (both sides are divided by 4)-4 = -8 : 2 ... 12 : 2 = ... (both sides are divided by 2)... = -8 : ... 12 : = ... (both sides are divided by )8 = -8 : -1 ... 12 : -1 = -12 (both sides are divided by -1)... = -8 : -2 ... 12 : -2 = ... (both sides are divided by -2)
  13. 13. Compare the sign in the box that you have filled with the signof the beginning statement. What happens if both sides aremultiplied by a positive number, by zero, or by negativenumber? And what happens if both sides are divided by apositive number , or by a negative number?Compare the sign in the box that you have filled with the signof the beginning statement. What happens if both sides aremultiplied by a positive number, by zero, or by negativenumber? And what happens if both sides are divided by apositive number , or by a negative number?-8 > -12... = -8 : 4 ... -12 : 4 = ... (both sides are divided by 4)-4 = -8 : 2 ... -12 : 2 = ... (both sides are divided by 2)... = -8 : ... -12 : = ... (both sides are divided by )8 = -8 : -1 ... -12 : -1 = 12 (both sides are divided by -1)... = -8 : -2 ... -12 : -2 = ... (both sides are divided by -2)
  14. 14. ProPerties of multiPlication ordivision on both sides of an inequalityFor an inequality :• If both sides are multiplied or divided by apositive number (non zero), then the sign ofthe inequality does not change.• If both sides are multiplied or divided by apositive number (non zero), then the sign ofthe inequality changes into the opposite.
  15. 15. Determine the solution of inequality 3x – 1 < x + 3 with x variable of whole number set!Method 1 :1.By changing the sign of “<“ with “=“, we get the following equation 3x – 1 = x + 3.2.Solve the equation.3x – 1 = x + 32x = 4x = 23.Take one counting number less than 2 and one counting number greater that 2.For example 1 and 3.4. Check which one meets 3x – 1 < x + 3.• x = 1 then 3x – 1 < x + 33.(1) – 1 < 1 + 32 < 4• x = 3 then 3x – 1 < x + 33.(3) – 1 < 3 + 38 < 6So, the solution set of 3x – 1 < x + 3 is {x | x < 2 ; x is a member of wholenumber}Method 1 :1.By changing the sign of “<“ with “=“, we get the following equation 3x – 1 = x + 3.2.Solve the equation.3x – 1 = x + 32x = 4x = 23.Take one counting number less than 2 and one counting number greater that 2.For example 1 and 3.4. Check which one meets 3x – 1 < x + 3.• x = 1 then 3x – 1 < x + 33.(1) – 1 < 1 + 32 < 4• x = 3 then 3x – 1 < x + 33.(3) – 1 < 3 + 38 < 6So, the solution set of 3x – 1 < x + 3 is {x | x < 2 ; x is a member of wholenumber}
  16. 16. Method 2 :3x – 1 < x + 3⇔ 3x – 1 + 1 < x + 3 + 1⇔ 3x < x + 4⇔ 3x + (-x) < x + (-x) + 4⇔ 2x < 4⇔ 2x : 2 < 4 : 2⇔ x < 2Because x is element of whole number, then the value of x which conforms with x < 2 are x = 0and x = 1 . So, the is solution set is {0,1}.To, determine equivalent equations, you can also perform the following methods.3x – 1 < x + 3⇔ 3x – 1 + (-3) < x + 3 + (-3)⇔ 3x -4 < x⇔ 3x + (-3x) - 4 < x + (-3x)⇔ -4 < -2x⇔ -4 : -2 < -2x : -2⇔ 2 > xSince x is element of whole number, then the value of x which conforms with x < 2 are x = 0and x = 1 . So, the is solution set is {0,1}. Using number line, chart of solution set is as shown infigure below.Both sides are added by 1Both sides are added by 1Both sides are added by -xBoth sides are added by -xBoth sides are divided by 2Both sides are divided by 2Both sides are added by -3Both sides are added by -3Both sides are added by -3xBoth sides are added by -3xBoth sides are divided by -2, butthe sign of the inequality ischanged to >Both sides are divided by -2, butthe sign of the inequality ischanged to >
  17. 17. Determine the solution of this inequality !Solution by using method I:⇔⇔⇔⇔⇔⇔⇔⇔⇔Both sides are multiplied by 6Both sides are multiplied by 6Both sides are subtractedd by 4Both sides are subtractedd by 4Both sides are subtractedd by 9xBoth sides are subtractedd by 9xSolving Inequality of Fraction FormSolving Inequality of Fraction Form232)2(31 xx +>+232)2(31 xx +>+)232(6)2(316xx +>+×xx 912)2(2 +>+xx 91242 +>+xx 9412442 +−>−+xx 982 +>xxxx 99892 −+>−87 >− x871771×−<−×− x78−<xBoth sides are multiplied byBoth sides are multiplied by 71−
  18. 18. Solution by using method II:⇔⇔⇔⇔⇔⇔⇔⇔This inequality can be solved by the other method232)2(31 xx +>+2323231 xx +>+32232323231−+>−+ xxxx2331131+>xxxx23233112331−+>−3116962>− xx31167>− x3476)67(76×−<−×− x78−<xBoth sides are multiplied byBoth sides are multiplied by 76−Both sides are subtracted byBoth sides are subtracted by32−Both sides are subtracted byBoth sides are subtracted by x23−

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