Answer:
- start by learning to do one-step <em>equations</em>
- learn the <em>ordering rule</em> for inequalities
Step-by-step explanation:
<u>Solving One-Step Equations</u>
A "one-step" equation is one that requires one step to obtain a solution. You probably could do these in 1st or 2nd grade, as soon as you began to learn arithmetic facts.
For example, one that requires an <u>addition step</u> is ...
x - 3 = 5
Since you know the addition fact ...
8 - 3 = 5 . . . . . . . . comparing to the above, we see x=8
you can probably solve this without even thinking about it. In Algebra, we learn to solve this sort of equation by "undoing" the subtraction of 3. That "undo" operation is the addition of 3 to both sides of the equation.
x -3 +3 = 5 +3
We can do this because the addition property of equality tells us that the value of the variable is unchanged if we add the same thing to both sides of the equation. (The quantity we add can be positive or negative.)
Simplifying the equation we just made, we get ...
x + 0 = 8
and the identity element property of addition tells us adding 0 doesn't change anything, so this is equivalent to ...
x = 8
In summary, for this equation, the "one step" was to add the opposite of the constant that is on the same side of the equal sign as the variable. We added that value to both sides of the equation. This gets the variable by itself, which is the goal.
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Similarly, there are equations where <u>the "one step" is multiplication</u> or division by some number. Such an equation might look like ...
3x = 6
Since you know your multiplication facts, you can probably solve this without too much thinking:
3·2 = 6 . . . . . . . . comparing to the above, we see x=2
In Algebra, we learn to solve this sort of equation by "undoing" the multiplication by 3. That "undo" operation is the division by 3 of both sides of the equation. We can also think of this as multiplying both sides by 1/3, the reciprocal of the coefficient of x.
(3x)/3 = 6/3
We can do this because the multiplication property of equality tells us that the value of the variable is unchanged if we multiply both sides of the equation by the same thing. (The quantity we add can be positive or negative.
Simplifying the equation we just made, we get ...
x·1 = 2
and the identity element property of multiplication tells us that multiplying by 1 doesn't change anything, so this is equivalent to ...
x = 2
In summary, for this equation, the "one step" was to multiply both sides of the equation by the reciprocal of the coefficient of the variable. This gets the variable by itself, which is the goal.
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<u>Solving One-Step Inequalities</u>
In an inequality, the equal sign is replaced by an ordering symbol, one of {<, >, ≤, ≥}. For addition by any number or multiplication by a positive number, an inequality is <em>solved exactly the same way as an equation</em>.
<em>Ordering Rule</em>
For multiplication (equivalently, division) by a <em>negative</em> number, the direction of the ordering must be reversed.
You can see this latter case if you consider the inequality ...
1 < 2
Now, when we multiply both sides of this by -1, we must reverse the symbol so we have ...
-1 > -2
This maintains the proper ordering of the two sides of the inequality.
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<u>Examples</u>:
-3x ≥ 6
x ≤ -2 . . . . . divide both sides by -3; reverse the ordering symbol
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x -3 > 5
x > 8 . . . . . . add 3 to both sides; same as for the equation above
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<em>Comment on where to start</em>
Whenever you approach the solution of any equation, you first identify the variable of interest, then the operations that are being done to it. You will want to undo those operations by making use of additive and multiplicative identity element rules: add the opposite to get a sum of zero; multiply by the reciprocal to get a product of 1.
Whatever <em>operations you perform must be done to both sides of the equation</em>. (Simplifying operations, such as using the distributive property or adding like terms, can be done to only one side of the equation.)
Be aware of the effect on ordering when you perform operations on inequalities.
