Download Solving Two-Step Equations

Solving Two-Step Equations
3-1
A two step equation contains two operations. To solve, use inverse operations
to undo each operation in reverse order, opposite the order of operations. First, undo
addition and or subtraction. Then, undo multiplication and or division.
Example 1
Example 2
Example 3
m
9 5
7
7c  4  32
5(f  2)  9  5f
Example 4
5d   9   24
Example 5
10  9  x
becomes
Example 6
The previous example
could be done like this.
9  x  10
Solving Equations Having Like Terms and Parentheses
3-2
In order to solve equations with like terms, you must first simplify the
equation by combining like terms, the same way you previously simplified
expressions. After the equation has been simplified, you solve using inverse
operations and begin working backwards, opposite the order of operations.
Example 1
Example 2
7c  8  2c  17
16  8r  4r  4  24
8r  4r  16  4  24
simplifies to this
7c  2c  8  17
simplifies to this
Example 3
6   y  42  2 y
 y  2 y  42  6
simplifies to this
If the question contains parentheses in addition to like terms that need to be
simplified, first use distributive property to get rid of the parentheses. Then, follow
the same process you used in examples one through three.
Example 4
12( x  3)  3 x  117
Example 5
20  14  3( x  8)
14  3( x  8)  20
12 x  36  3x  117
12 x  3x  36  117
14  3 x  24  20
3 x  14  24  20
Find the value of x.
Example 7
since
2( x  7)  2(4)  2 x  14  8
 2 x  22
or
Perimeter = 48 units
4
x+7
Example 6
7 x  (10  x)  58
x74 x74
x x7474
2 x  22
then
2 x  22  48
 22  22
2x
26

2
2
x
 13 units
Solving Equations with Variables on Both Sides
3-3
To solve equations with variables on each side, use the Addition or Subtraction
Property of Equality to write an equivalent equation with the variables on one side of
the equation. Then, solve the equation.
Example 1
2y  7 
y
Example 2
2h  16  3h  6
Example 3
28  4d  5d  17
Example 4
Twice a number is 60 more than five times the number. What is the number?
2x  60  5x
Example 5
Eight less than three times a number equals the number. What is the number?
3x  8  x
3-3 continued
Equations often contain grouping symbols such as parenthesis or brackets.
The first step in solving these equations is to use the Distributive Property to remove
the grouping symbols.
Example 6
4(m  7)  12
Example 7
7(2p  3)  8  6p  29
4m  28  12
Example 8
Find the perimeter of the square.
Since this is a square, the sides are equivalent.
3x – 1
3x  1  2x  4
2x + 4
So, the length of one side would be 3x  1
3 5  1
15  1
14
and the perimeter would be
14
4
96 units
3-3 continued
Some equations have no solution. That is, there is no value for the variable
that would result in a true sentence. For such an equation, the solution set is called
the null or empty set, and is represented by the symbol  or   .
Example 9
5(f  2)  9  5f
Ten never equals nine. This is NEVER true.
So the solution is  .
Other equations may have every number as the solution. An equation that is
true for every value of the variable is called an identity.
Example 10
2(x  2)  3  2x  1
Since -1 ALWAYS equals -1, the solution is all numbers. In
other words, it doesn’t matter what number you use for x,
any number would result in a true statement.
Solving Inequalities Using Addition or Subtraction
3-4
A mathematical sentence that contains the symbols < (less than), > (greater
than),  (less than or equal to), or  (greater than or equal to) between two
expressions is called an inequality. For example, the statement that the speed limit is
45mph can be shown by the sentence s  45, which implies you can go 45 miles per
hour or slower. Inequalities with variables are called open sentences. The solution of
an inequality with a variable is the set of all numbers that the variable could be
replaced with that would result in a true statement.
* Hint: When writing a verbal description as an inequality, keep in mind the
following; when you see these phrases, use these symbols

no more than or at most
no less than or at least

The solutions of inequalities can be graphed on a number line. For example,
a > 4 or 4 < a
a < 4 or 4 > a
open circle and darken to the right
a  4 or 4  a
open circle and darken to the left
a  4 or 4  a
closed circle and darken to the right
closed circle and darken to the left
Some inequalities can be solved by using the Addition and Subtraction
Properties of Inequalities. These properties say that when you add or subtract the
same number from each side of an inequality, the inequality remains true.
Example 1
n 3  6
Example 2
2m
5
Example 3
10  x  5
3-4 continued
Example 4
d  0.15  4
Since the solutions to an inequality include all rational numbers that satisfy it,
inequalities have an infinite number of solutions.
Remember that the solutions for an inequality can also be graphed.
Example 5
This is the graph of the solution from example 2 above;
m  7
Example 6
This is the graph of the solution from example 4 above;
d   3.85
3–5
Solving Inequalities Using Multiplication or Division
Some inequalities can be solved by using the Multiplication and Division
Properties of Inequalities. These properties say that when multiplying or dividing
each side of an inequality by the same positive number, the inequality remains true.
In such cases, the inequality symbol does not change.
Example 1
81  9d
Example 2
p
 12
3
Example 3
m
 0.5
3
When multiplying or dividing each side of an inequality by a negative
number, the inequality symbol must be reversed.
Example 4
81  9d
Example 5
p
  12
3
Example 6
m
 0.5
3
3 – 5 continued
Example 7
1
 x  9
3
Example 8
y
 19 
0.3
Write the verbal sentence as an inequality. Then solve the inequality.
Example 9
16 is greater than twice a number.
16  2c
Example 10
The quotient of a number and – 4 is no more than 32.
d
 32
4
Solving Multistep Inequalities
3–6
When solving inequalities that involve more than one operation, work
backward to undo the operations, just as when you solve multi-step equations.
Remember to reverse the inequality symbol if you multiply or divide each side
of an inequality by a negative number, as in example 2.
Example 1
2x  12  12
Example 2
5  4k  21
Sometimes you will need to use the Distributive Property to begin simplifying
inequalities that contain grouping symbols before you solve.
Example 3
3  x  3  7.5
3 – 6 continued
In some questions you will have to begin by eliminating a division step by
multiplying by the divisor.
Example 4
18  n
6
2