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One Step Inequalities

Solving One Step Inequalities

Tutorials you should read before this one:  Graphing Inequalities, Solving One-Step Equations

A brief explanation
Solving inequalities works the same way as solving equations, but with two differences.  First, students must know how to graph their solutions on a number line.  Second is the golden rule of inequalties, if you multiply/divide by a negative during your solving process, you have to flip the sign.  Be careful with your graphing, though!  Some of those memory tricks you  have relied on in the past may actually cause you to make mistakes when graphing.  I'll point out some of these common mistakes here.

Why do we have to graph an inequality solution?
When we solve an equation, such as x + 2 = 5, there is only one possible solution:  x = 3.  The only value that will make this equation true is the number 3. But when it comes to an inequality, there is no possible way we can write down every solution.  For instance, the problem x + 1 > 2 has infinite solutions; x can be 3, 4, 5, 6 and so on.  And notice that in our incomplete list, we didn't include all those other possible solutions such as 1.23 or 4.000011.  Since we can't write a finite list, we use the number line to show those solutions graphically.  

Solving and Graphing One Step Inequalities
Remember, the steps for solving an inequality is the same as solving equations.  Each inequality problem should be looked at in regards to the two sides of a problem, the left side and the right side separated by an inequality symbol.  Our goal is get one single, positive variable by itself on one side of our inequality.  

Step-by-Step Examples
EXAMPLE:
x + 7 > 11The variable x is on the left side, so the only term I want on that left side is the x.  But I have two terms on the left side:  x and 7.  In order to get x by itself, I need to get rid of the +7,
x  +  7  >  11
      -7       -7
We get rid of terms we don't want by adding the opposite.  So we will add a -7 to both sides.  Now combine like terms.
x > 4  The +7 and -7 will cancel out, which is why we chose -7 to add to both sides in the first place.  On the right side, 11 + -7 = 4.  At this point, we have a single positive x and therefore have reached our final answer.  Now we have to graph our answer.
x > 4The first step is to think about how to indicate our end point.  In this case, our end number is 4.  Since our symbol does not have the 'equal to' option, we know we'll use an open circle.
x > 4Make sure you read your symbol, starting with the variable.  Remember, if you get to the "mouth" first, it's greater than.  If you get to the "tail," then it's less than.  So our solution x > 4 reads as x greater than 4.  Greater than shades to the right of our endpoint.
   
TIP
Next is an example of the type of problem that gives students the most difficulty, unless you remember to always read your inequality starting from the variable.

EXAMPLE:  
-5 >= x - 4
The variable x is on the right side, so all I want on that side is the x term.
I need to get rid of that -4.  
-5 >= x - 4
  +4         +4
To get rid of the -4, we will add a 4 to both sides.  Now combine like terms.
-1 >= xThe -4 and 4 will cancel out on the right side.  On the left side, -5 + 4 = -1.  At this point, we have a single positive x and therefore have reached our final answer.  Now let's graph it.
-1 >= xOur end number is -1.  Since this has the 'equal to' option, we know we'll use a closed circle.
-1 >= xTo always graph correctly, you must read your inequality starting from the variable.  In other words, since our variable is on the right, we read from right to left.  We start with the x, and we see that the "tail" of our inequality is facing our variable, so it's "less than or equal to."  
All together:  x is less than or equal to -1.  
-1 >= xLess than shades to the left.  


EXAMPLE:
-6x < 24 This time our variable is on the left side.  There is only one term on that left side which means there are no extra terms to get rid of.  
-6x/-6 < 24/-6
Remember, when there is only one term left, this is where we use the inverse operations multiplication/division to get rid of unwanted coefficients.  In this case, since we have -6x (which means x is getting multiplied by -6), we do the opposite operation which is divide by -6 on both sides.
x > -4  On the left side we have one single positive x since -6/-6 = 1.  On the right side, 24/-6 = -4.  This time to solve our inequality, we had to divide both sides by -6.  Remember, whenever you multiply or divide by a negative in the solving process, you have to flip your symbol.
x > -4Our solution reads as x is greater than -4.  Our end number is -4, open circle since our symbol does not have the equal to option.
x > -4Greater than shades to the right.


EXAMPLE:
-1 < x / -2
This time our variable is on the right side.  There is only one term on the right side which means there are no other terms to get rid of.  
-2(-1) < (x/-2)(-2)
When there is only one term left, this is where we use the inverse operations multiplication/division to get rid of unwanted coefficients.  In this case, since our term is x/-2 (which means x is getting divided by -2), we do the opposite operation which is multiply by -2 on both sides.
2 > x  The -2(-1) = 2 on the left, and on the right, -2/-2 is 1 so we have one single positive x and have reached our answer.  But remember, whenever you multiply or divide by a negative in the solving process, you have to flip your symbol.
2 > x
Always read starting from the variable.  In this case, we have to read right to left.  Our solution reads as x is less than 2 since as we move left from x, we reach the "tail" first.  Our endpoint is 2, open circle since our symbol does not have the equal to option.
2 > x
Less than shades to the left.


ONLINE PRACTICE

Play Who Wants to be a Millionaire with One-Step Inequalities
Practice your one step equation solving here.