GCSE Linear Inequalities jamiedrfrostmaths com www drfrostmaths com

GCSE : : Linear Inequalities jamie@drfrostmaths. com www. drfrostmaths. com @Dr. Frost. Maths Last modified: 26 th March 2021

www. drfrostmaths. com Everything is completely free. Why not register? Register now to interactively practise questions on this topic, including past paper questions and extension questions (including UKMT). Teachers: you can create student accounts (or students can register themselves), to set work, monitor progress and even create worksheets. With questions by: Teaching videos with topic tests to check understanding. Questions organised by topic, difficulty and past paper.

Further Resources on this Topic Key Skills allow repetitive practice of randomly generated questions, with short worked example videos and detailed feedback. Teachers can also use these to produce randomly-generated worksheets. Simply type the ‘K___’ into the Dr. Frost. Maths search bar or use the direct link. Key Skill Direct Link K 252: Represent an inequality on a number line. www. drfrostmaths. com/keyskills. php? ksid=252 K 253: Interpret an inequality represented on a number line. www. drfrostmaths. com/keyskills. php? ksid=253 K 254: Solve a linear inequality. www. drfrostmaths. com/keyskills. php? ksid=254 Exam Skills involve broader topic areas and consist of past paper exam questions (as well as user contributions). Teachers can download teaching resources and browse exam questions, and students can practise questions online. Exam Skill Direct Link E 95: Represent solutions of an inequality on a number line. www. drfrostmaths. com/resourceexplorer. php? tid=44&skid=95 E 93: Solve linear inequalities. www. drfrostmaths. com/resourceexplorer. php? tid=44&skid=93

Inequalities in everyday life Where in real life might we use phrases like “at least”, “more than”, “less than” and “at most”? Don’t worry about this yet – we’ll cover it in a second. Real-life scenario How we could represent mathematically “You can have at most 20 ? people at your party. ” ? “I was chased by at least 10 ? zombies!” ? I’ll visit next in less than a ? month. ” ? “My cat’s IQ is between 120 ? and 140. ” ?

Why we need inequalities in maths Inequalities are needed in mathematics when we need to represent a range of values. Equation: Number of Solutions: ? ? 0 solutions ? ? A ‘range’ of values often involves infinitely possible many values. So we need inequalities to be able to represent them, as it’s not possible to list all the values.

Reading inequalities ! Notice the symbol is taller on the side which is larger. This doesn’t include 7 itself. What it means ? This does include 7. ? ? ?

Further Examples Are the following inequalities true or false? True False In words: “ 3 is less than 4”. This is true: 3 is a smaller value than 4. In words: “-5 is greater than 1”. This is not true: -5 is not the larger value. In words: “ 5 is less than or equal to 5”. This is true: the left can either be less than or equal to the right. 5 is equal to 5!

Test Your Understanding So Far Write the following as inequalities: 1 2 3 4 5 6 ? ? ?

Inequalities on Number Lines We can use a filled circle on a number line to indicate we want to include the value. But what about: We again use a filled circle to indicate that we want to include 4. ? But we also have an arrow pointing left to say we also want any value less than 4.

Further Examples We again want to include 1, but our arrow is right this time to indicate values greater than 1. ? We again have an arrow left to indicate “less than 2”, but this time we DON’T want to include 2 itself. We use an unfilled circle to indicate that 2 is excluded. ?

Two-ended inequalities ? We therefore tend to write inequalities in this form when we want to say a variable is between two values. On a number line… ? ?

Test Your Understanding ? Inequality 1 2 ? Inequality 3 4 ? Inequality

Exercise 1 Available on printable worksheet 3 Write inequalities that represent 1 ? ? ? a b c d e f g 2 a b c d e ? ? ? the following number lines. ? a ? b ? c ? d ? e f ? ?

Exercise 1 Available on printable worksheet 4 Represent the following inequalities on a number line. a ? b ? c ? d ? e ? f ?

Solving Linear Inequalities behave in a similar way to equations: whatever we do to one side of the equation, we have to do the same to the other. Example values… ? +1 +1 ? Example values… ? ?

Examples ? ?

Check Your Understanding So Far 1 ? 2 ? 3 ?

Variable on both sides ? ? ?

Check Your Understanding ?

We might think that we can divide both sides of the equation by a negative number to solve: Think of a number that works with this inequality. e. g. ? 1 ? ! When you divide or multiply both sides of an inequality by a negative number, reverse the direction of the inequality. Does it work with the simplified inequality? But it’s probably easiest to avoid needing to divide by a negative number in the first place…

Example Method 1: Dividing by a negative number ? Method 2: Put the variable term on the side where it’ll be positive. ? This method avoids the need to remember to reverse the direction of the inequality, as we’re dividing by a positive number.

Test Your Understanding ? ?

Solving double-ended inequalities Key Point: Do the same operation to all 3 ‘parts’ of the inequality.

Test Your Understanding ?

Exercise 2 Available on printable worksheet 3 1 a b c d e f g h i j 2 a b c d e f g h ? ? ? ? a b c d e f g 4 a b c d e N a b ? ? c ? ? ? ?

Combining Inequalities We’ve already seen examples where we’ve combined inequalities together: Discussing in pairs, think about how we could combine the following sets of information: Alice says: “Charles is at least 12 years old”. Bob says “Charles is at least 15 years old”. Shota says: “Tasmin is between 10 and 14”. Fleur says “Tasmin is at least 13”. What do we know about Charles’ age? What do we know about Tasmin’s age? This is equivalent to just saying Charles is at least 15 years old. If he was at least 15, then he would have to be at least 12 anyway, so Alice’s statement is redundant. ? Tasmin must be between 13 and 14 years old. ?

Combining Inequalities using Number Lines “Charles is at least 12 years old. ” “Charles is at least 15 years old. ” To combine these together, place your finger vertically up the page. We will gradually ‘scan’ our finger from left to right. At 12, our finger is over the top line, but not the bottom. So 12 is not in our combined inequality. However at 15, we’re on both lines (recall the filled circle means 15 is included). Values will only appear on our combined inequality if our finger is on BOTH lines. At the moment our finger is on neither. And for any value above 15, our finger is still over both lines. So values above 15 are in our combined inequality.

Further Examples Combined At 15, the top line is included, but the second one isn’t (because an unfilled circle indicates the value is not included). We therefore don’t include the value in the combined inequality (as an unfilled circle). ? Between 15 and 17, we’re on both lines, so this region will be included in our combined inequality. 17 is on both lines, because the circle at the top is filled. Above 17 we’re only on the second line, so we don’t include.

Harder Example Combined ? ?

Test Your Understanding 1 Combined ? ? ? ? 2 Combined ? ? ? ?

Solving Harder Double-Ended Inequalities Split into two separate inequalities. Solve each separately. Combine together. ?

Exercise 3 Available on printable worksheet 1 ? ? ? ? a b c d e f g h i j 2 a b c d e f ? ? ?