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Luton Dunstable. Manchester. Middlesbrough. Newcastle Gateshead. Norwich. Nottingham. Command And Conquer Generals Zero Hour Pc. Nuneaton Bedworth. Oldham. Perth. Peterborough. Portsmouth. Rochdale. Rugby. Rugeley. Salford. Sandwell. Sheffield. Solihull. Stafford. Stirling. Stockport. Stoke on Trent Newcastle. Stratford upon Avon. Sunderland. Swindon. Tameside. Tamworth. Trafford. Wakefield District. Walsall. Warrington. Weymouth Portland. Whitburn. Wigan Borough. Wolverhampton. York. Solving Inequalities. Sometimes we need to solve Inequalities like these Symbol. Words. Example greater thanx 3 2lt less than. Solving. Our aim is to have x or whatever the variable is on its own on the left of the inequality sign Something like x lt 5or y 1. We call that solved. Example x 2 1. Subtract 2 from both sides x 2 2 1. Simplify x 1. Solved How to Solve. Solving inequalities is very like solving equations. Direction Which way the arrow pointsSome things can change the direction Safe Things To Do. These things do not affect the direction of the inequality Add or subtract a number from both sides. Multiply or divide both sides by a positive number. Simplify a side. Example 3x lt 73. We can simplify 73 without affecting the inequality 3x lt 1. But these things do change the direction of the inequality lt becomes for example Multiply or divide both sides by a negative number. Swapping left and right hand sides. Example 2y7 lt 1. When we swap the left and right hand sides, we must also change the direction of the inequality 1. Here are the details Adding or Subtracting a Value. We can often solve inequalities by adding or subtracting a number from both sides just as in Introduction to Algebra, like this Solve x 3 lt 7. If we subtract 3 from both sides, we get x 3 3 lt 7 3 x lt 4. And that is our solution x lt 4. In other words, x can be any value less than 4. What did we do We went from this To this x3 lt 7x lt 4 And that works well for adding and subtracting, because if we add or subtract the same amount from both sides, it does not affect the inequality. Example Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy. What If I Solve It, But x Is On The Right No matter, just swap sides, but reverse the sign so it still points at the correct value Example 1. If we subtract 5 from both sides, we get 1. That is a solutionBut it is normal to put x on the left hand side Do you see how the inequality sign still points at the smaller value 7 And that is our solution x 7. Note x can be on the right, but people usually like to see it on the left hand side. Multiplying or Dividing by a Value. Another thing we do is multiply or divide both sides by a value just as in Algebra Multiplying. But we need to be a bit more careful as you will see. Positive Values. Everything is fine if we want to multiply or divide by a positive number Solve 3y lt 1. If we divide both sides by 3 we get 3y3 lt 1. And that is our solution y lt 5. Negative Values. Why Well, just look at the number line For example, from 3 to 7 is an increase, but from 3 to 7 is a decrease. See how the inequality sign reverses from lt to Let us try an example Solve 2y lt 8. Let us divide both sides by 2. And that is the correct solution y 4Note that I reversed the inequality on the same line I divided by the negative number. So, just remember When multiplying or dividing by a negative number, reverse the inequality. Multiplying or Dividing by Variables. Here is another tricky example Solve bx lt 3b. It seems easy just to divide both sides by b, which gives us x lt 3. But we dont know if b is positive or negative, so we cant answer this one To help you understand, imagine replacing b with 1 or 1 in the example of bx lt 3b if b is 1, then the answer is x lt 3 but if b is 1, then we are solving x lt 3, and the answer is x 3. The answer could be x lt 3 or x 3 and we cant choose because we dont know b. So. Do not try dividing by a variable to solve an inequality unless you know the variable is always positive, or always negative. A Bigger Example. Solve x 32 lt 5. First, let us clear out the 2 by multiplying both sides by 2. Because we are multiplying by a positive number, the inequalities will not change. Now add 3 to both sides x 3 3 lt 1. And that is our solution x lt 7. Two Inequalities At Once How do we solve something with two inequalities at once Solve 2 lt 6 2x3 lt 4. First, let us clear out the 3 by multiplying each part by 3 Because we are multiplying by a positive number, the inequalities will not change. Now subtract 6 from each part 1. Now multiply each part by 12. Because we are multiplying by a negative number, the inequalities change direction. Install Ntop Centos 7. And that is the solutionBut to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over and make sure the inequalities point correctly 3 lt x lt 6. Summary. Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own. But these things will change direction of the inequality. Multiplying or dividing both sides by a negative number. Swapping left and right hand sidesDont multiply or divide by a variable unless you know it is always positive or always negative.