Solving Equations and Inequalities
• The basic principle in solving equations and inequalities is that you can
manipulate them in any way as long as you do the same thing to both sides. For example you may add a number to both sides, or you may divide or multiply both sides with same number etc.
• By using the following six-step method, you can solve most of the
equations and inequalities. The method is explained with the help of an
example.
Example:
if
1/2.x +3(x-2)=2(x+1)+1, what is the value of
x?
Step 1:
Get rid of fractions and decimals by multiplying both sides by the LCD.
Multiply each side by
2 to get
x+6(x-2)=4(x+1)+2 .
Step 2:
Get rid of all parentheses by solving them i.e.
x+6x-12 = 4x+4+2
Step 3:
Combine like terms on each side.
7x -12 = 4x + 6
Step 4:
By adding and subtracting get all the variables on one side (mostly left).
Subtract
4x from each side to get:
3x - 12 = 6
Step 5:
By adding or subtracting get all plain numbers on the other side.
Add
12 to each side to get:
3x = 18
Step 6:
Divide both sides by the coefficient of the variable.(If you are dealing with an inequality and you divide with a negative number, remember to reverse the inequality.) i.e. Divide both sides by
3 to get:
x = 6
• When you have to solve one variable and the equation/inequality involve
more than one variable, treat all other variables as plain numbers and
apply the six-step method.
Example:
if
a=3b-c what is the value of
b in terms of
a and
c?
Solution:
Step 1: There are no fractions and decimals.
Step 2: There are no parentheses.
Step 3: There are no like terms.
Step 4: By adding and subtracting get all the variables on one side i.e. Remember there is only one variable b, which is on one side only.
Step 5: By adding or subtracting get all plain numbers on the other side. Remember we are considering a and c as plain number. Add c to
each side to get:
a+c=3b
Step 6: Divide both sides by the coefficient of the variable.
Divide both sides by
3 to get:
a+c/3 = b
• It is not necessary to follow these steps in the order specified. Some times it makes the problem much easier, if you change the order of these steps.
• Doing the same thing on each side of an equation does not mean doing the
same thing to each term of the equation. This is very important if you are
doing divisions, or dealing with exponents and roots.
• Another type of equation is that in which the variable appears in exponent.
These equations are basically solved by inception.
If
2 ^ x+32 =32 ( ^ stands for exponent means raise to power), what is the value of
3^x+2?
Solution:
2^x+3=32 ⇒
2^x+3=2^5 ⇒
x+3=5 ⇒
x=2
Now as
x=2 you can get
x=2 ⇒
x+2=4 ⇒
3^x+2 = 3 ^4 =81
• A system of equations is a set of two or more equations having two or more variables. To solve such equations, you must find the value of each
variable that will make each equation true.
• To solve a system of equations, add or subtract them to get a third
equation. If there are more than two equations you can just add them.
Example:
If
x+y=10 and
x-y=10 then what is the value of
y?
Solution:
Add these two equations:
(x+y=10) + (x-y=10) gives us
2x=12 i.e. x=6
Now replacing
x with
6 in the first equation :
6+y=10 gives us
y=4
• If you know the value of one variable in a system of two equations, you can use this value to get the value of the other variable. As it is done in the
previous question.
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