Sometimes we need to solve Inequalities like these: Show
SolvingOur aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:
We call that "solved". Example: x + 2 > 12Subtract 2 from both sides: x + 2 − 2 > 12 − 2 Simplify: x > 10 Solved! How to SolveSolving inequalities is very like solving equations ... we do most of the same things ... ... but we must also pay attention to the direction of the inequality.
Some things can change the direction! < becomes > > becomes < ≤ becomes ≥ ≥ becomes ≤ Safe Things To DoThese things do not affect the direction of the inequality:
Example: 3x < 7+3We can simplify 7+3 without affecting the inequality: 3x < 10 But these things do change the direction of the inequality ("<" becomes ">" for example):
Example: 2y+7 < 12When we swap the left and right hand sides, we must also change the direction of the inequality: 12 > 2y+7 Here are the details: Adding or Subtracting a ValueWe can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this: Example: x + 3 < 7If we subtract 3 from both sides, we get: x + 3 − 3 < 7 − 3 x < 4 And that is our solution: x < 4 In other words, x can be any value less than 4. What did we do?
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: 12 < x + 5If we subtract 5 from both sides, we get: 12 − 5 < x + 5 − 5 7 < x That is a solution! But it is normal to put "x" on the left hand side ... ... so let us flip sides (and the inequality sign!): x > 7 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 ValueAnother 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 ValuesEverything is fine if we want to multiply or divide by a positive number: Example: 3y < 15If we divide both sides by 3 we get: 3y/3 < 15/3 y < 5 And that is our solution: y < 5 Negative Values
Why?Well, just look at the number line! For example, from 3 to 7 is an increase,
See how the inequality sign reverses (from < to >) ? Let us try an example: Example: −2y < −8Let us divide both sides by −2 ... and reverse the inequality! −2y< −8 −2y/−2 > −8/−2 y > 4 And that is the correct solution: y > 4 (Note 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 VariablesHere is another (tricky!) example: Example: bx < 3bIt seems easy just to divide both sides by b, which gives us: x < 3 ... but wait ... if b is negative we need to reverse the inequality like this: x > 3 But we don't know if b is positive or negative, so we can't answer this one! To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:
The answer could be x < 3 or x > 3 and we can't choose because we don't 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 ExampleExample: x−32 < −5First, 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. x−32 ×2 < −5 ×2 x−3 < −10 Now add 3 to both sides: x−3 + 3 < −10 + 3 x < −7 And that is our solution: x < −7 Two Inequalities At Once!How do we solve something with two inequalities at once? Example: −2 < 6−2x3 < 4First, let us clear out the "/3" by multiplying each part by 3. Because we are multiplying by a positive number, the inequalities don't change: −6 < 6−2x < 12 Now subtract 6 from each part: −12 < −2x < 6 Now divide each part by 2 (a positive number, so again the inequalities don't change): −6 < −x < 3 Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction. 6 > x > −3 And that is the solution! But 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 < x < 6 Summary
What are the differences and similarities between equations and inequalities?Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.
How is solving a linear inequality is similar to solving a linear equation?Solving a system of linear inequalities is similar to solving system of linear equations but with inequalities we are not finding a point (or points) of intersect. Instead the solution set will be the region that satisfies all of the linear inequalities.
How is an inequality like an equation?*Solving inequalities is just like solving equations, use opposite operations to isolate the variable. *When multiplying or dividing by a negative number, the inequality sign must be reversed. You can represent the solution of an inequality in one variable on a number line.
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