GCSE: Linear Inequalities Skipton Girls High School Objectives: Solving linear inequalities, combining inequalities and representing solutions on number lines.

Writing inequalities and drawing number lines You need to be able to sketch equalities and strict inequalities on a number line. This is known as a ‘strict’ inequality. x > 3 x < -1 Means: x is (strictly) greater than 3. ? Means: x is (strictly) less than -1. ? 0 1 2 3 4 5 -3 -2 -1 0 1 2 ? ? x ≥ 4 x ≤ 5 Means: x is greater than or equal to 4. ? Means: x is less than or equal to 5. ? 2 3 4 5 6 7 2 3 4 5 6 7 ? ?

𝒙>𝟑 𝒙−𝟏>𝟐 Deal or No Deal?   We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 𝒙>𝟑 Can we add or subtract to both sides? 𝒙−𝟏>𝟐 Click to Deal   Click to No Deal

𝟐𝒙>𝟔 𝒙>𝟑 Deal or No Deal?   We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 𝟐𝒙>𝟔 Can we divide both sides by a positive number? 𝒙>𝟑 Click to Deal   Click to No Deal

𝒙<𝟏 𝟒𝒙<𝟒 Deal or No Deal?   We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 𝒙<𝟏 Can we multiply both sides by a positive number? 𝟒𝒙<𝟒 Click to Deal   Click to No Deal

𝒙<𝟏 −𝒙<−𝟏 Deal or No Deal?   We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 𝒙<𝟏 Can we multiply both sides by a negative number? −𝒙<−𝟏 Click to Deal   Click to No Deal

-4 -2 2 < 4 ‘Flipping’ the inequality × (-1) × (-1) If we multiply or divide both sides of the inequality by a negative number, the inequality ‘flips’! OMG magic! -4 -2 2 < 4 × (-1) Click to start Bro-manimation × (-1)

−𝑥<3 −3<𝑥 𝑥>−3 1−3𝑥≥7 1−7≥3𝑥 −6≥3𝑥 −2≥𝑥 𝑥≤−2 Alternative Approach Or you could simply avoid dividing by a negative number at all by moving the variable to the side that is positive. −𝑥<3 −3<𝑥 𝑥>−3 1−3𝑥≥7 1−7≥3𝑥 −6≥3𝑥 −2≥𝑥 𝑥≤−2 ? ? ? ? ? ?

2𝑥<4 𝑥<2 𝑥<3 −𝑥>−3 4𝑥≥12 𝑥≥3 −4𝑥>4 𝑥<−1 − 𝑥 2 ≤1 Quickfire Examples Solve 2𝑥<4 𝑥<2 ? Solve 𝑥<3 −𝑥>−3 ? Solve 4𝑥≥12 𝑥≥3 ? Solve −4𝑥>4 𝑥<−1 ? − 𝑥 2 ≤1 Solve 𝑥≥−2 ?

1 𝑥 <2 1<2𝑥 Deal or No Deal?   We can manipulate inequalities in various ways, but which of these are allowed and not allowed? 1 𝑥 <2 Can we multiply both sides by a variable? 1<2𝑥 Click to Deal  Click to No Deal  The problem is, we don’t know if the variable has a positive or negative value, so negative solutions would flip it and positive ones wouldn’t. You won’t have to solve questions like this until Further Maths A Level!

3𝑥−4<20 𝑥<8 4𝑥+7>35 𝑥>7 5+ 𝑥 2 ≥−2 𝑥≥−14 7−3𝑥>4 𝑥<1 More Examples Hint: Do the addition/subtraction before you do the multiplication/division. Solve 3𝑥−4<20 𝑥<8 ? Solve 4𝑥+7>35 𝑥>7 ? Solve 5+ 𝑥 2 ≥−2 𝑥≥−14 ? Solve 7−3𝑥>4 𝑥<1 ? Solve 6− 𝑥 3 ≤1 𝑥≥15 ?

Click to start bromanimation Dealing with multiple inequalities Hint: Do the addition/subtraction before you do the multiplication/division. 8 < 5x - 2 ≤ 23 8 < 5x - 2 5x - 2 ≤ 23 and 2 < x and x ≤ 5 2 < x x ≤ 5 𝟐<𝒙≤𝟓 Click to start bromanimation

−𝟏<𝒙<𝟏 −𝟒<𝒙<𝟐 More Examples Hint: Do the addition/subtraction before you do the multiplication/division. Solve 𝟏<𝟐𝒙+𝟑<𝟓 −𝟏<𝒙<𝟏 ? Solve −𝟐<−𝒙<𝟒 ? −𝟒<𝒙<𝟐

𝟓<𝒙<𝟕 −𝟐<𝒙<𝟎 𝟏𝟏<𝟑𝒙−𝟒<𝟏𝟕 𝟏<𝟏−𝟐𝒙<𝟓 Test Your Understanding Solve 𝟓<𝒙<𝟕 𝟏𝟏<𝟑𝒙−𝟒<𝟏𝟕 ? Solve 𝟏<𝟏−𝟐𝒙<𝟓 −𝟐<𝒙<𝟎 ?

Exercise 1 Solve the following inequalities, and illustrate each on a number line: Sketch the graphs for 𝑦= 1 𝑥 and 𝑦=1. Hence solve 1 𝑥 >1 0 < x < 1 N1 2𝑥−1>5 𝒙>𝟑 −2𝑥<4 𝒙>−𝟐 5𝑥−2≤3𝑥+4 𝒙≤𝟑 𝑥 4 +1≥6 𝒙≥𝟐𝟎 𝑦 6 −1≤7 𝒚≤𝟒𝟖 1−𝑦 2 ≤𝑦 𝒚≥ 𝟏 𝟑 1−4𝑥>5 𝒙<−𝟏 5≤2𝑥−1<9 𝟑≤𝒙<𝟓 5≤1−2𝑥<9 −𝟒<𝒙≤−𝟐 10+𝑥<4𝑥+1<33 𝟑<𝒙<𝟖 1−3𝑥<2−2𝑥<3−𝑥 𝒙>−𝟏 1 ? 2 ? ? 3 ? You can get around the problem of multiplying/dividing both sides by an expression involving a variable, by separately considering when the denominator positive, and when it’s negative, and putting this together. Hence solve: 3 𝑥+2 >4 If we assume 𝒙+𝟐 is positive, then 𝒙>−𝟐 and solving gives 𝒙<− 𝟓 𝟒 . Thus −𝟐<𝒙<− 𝟓 𝟒 as we had to assume 𝒙>−𝟐. If 𝒙<−𝟐 then this solves to 𝒙>− 𝟓 𝟒 which is a contradiction. Thus −𝟐<𝒙<− 𝟓 𝟒 N2 4 ? 5 ? ? 6 7 ? 8 ? ? 9 ? 10 ? 11 ?

x ≥ 2 and x < 4 x < -1 or x > 3 x ≥ 2, x < 4 2 ≤ x < 4 Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. AND How would we express “x is greater than or equal to 2, and less than 4”? OR How would we express “x is less than -1, or greater than 3”? x ≥ 2 and x < 4 x < -1 or x > 3 ? ? x ≥ 2, x < 4 ? This is the only way you would write this – you must use the word ‘or’. 2 ≤ x < 4 ? This last one emphasises the fact that x is between 2 and 4.

Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. 2 ≤ x < 4 x < -1 or x > 4 0 1 2 3 4 5 -1 0 1 2 3 4 ? ?

x ≥ 2 and x < 4 x < -1 or x > 4 Combining inequalities It’s absolutely crucial that you distinguish between the words ‘and’ and ‘or’ when constraining the values of a variable. To illustrate the difference, what happens when we switch them? or and x ≥ 2 and x < 4 x < -1 or x > 4 0 1 2 3 4 5 -1 0 1 2 3 4 ? ?

7 > 𝑥 > 4 4>𝑥<8 4<𝑥>7 I will shoot you if I see any of these… 4>𝑥<8 This is technically equivalent to: x < 4 ? 4<𝑥>7 This is technically equivalent to: x > 7 ? 7 > 𝑥 > 4 The least offensive of the three, but should be written: 4 < x < 7 ?

Combining Inequalities In general, we can combine inequalities either by common sense, or using number lines... 2 5 Where are you on both lines?   4   Combined ? 𝒙>𝟓 2 5 2<𝑥<5 4 𝑥<4 ? Combined 𝟐<𝒙<𝟒

Test Your Understanding ? ? -1 5 1st condition ? -3 3 2nd condition ? Combined

Exercise 2 By sketching the number lines or otherwise, combine the following inequalities. 1 ? 2 ? 3 ? 4 ? ? 5 6 ? 7 ? 8 ? 9 ? 10 ? ? 11 12 ? 1 13 ? 14 ? 15 ? 2 9