Solving Equations with Fractions
It is best to clear a linear equation that contains fractions or decimals before proceeding
with the rest of the solution. Otherwise, the rest of the solution may require one to find
common denominators and convert numbers to equivalent fractions. One might have to
perform addition, subtraction, multiplication, and division with decimals rather than
integers. Integers are easier to use.
A linear equation containing fractions can be cleared by multiplying the whole
equation through with the least common denominator of any fractions present.
For example, consider the equation:
1/4x – 5/6 = 1/8x + 1/2
The denominators are 4, 6, 8, and 2. The least common denominators of these would
be 24. By multiplying the equation through by 24, an equivalent equation with allinteger coefficients is found to be the following:
6x – 20 = 3x + 12
If the original equation contains decimals, a slightly different technique is used.
Consider the equation:
3.16x + 1.1 = 2.24x – 2.00
To clear the equation of decimals, determine the maximum number of decimal places
in any of the numbers. The number of decimal places is 2. Use the power of 10 with
that many zeros to multiply the equation through; in this case, it would be the
following:
100(3.16x + 1.1 = 2.24x – 2.00), and the equivalent, easier-to-solve version of the
equation would be 316x + 110 = 224x – 200.
Formulas
Formulas are often used in everyday life. For example, formulas are used when finding
the volume of a can or the interest earned on a loan. While linear equations contain one
variable, such as 2x + 3 = 5, a formula usually contains more than one variable. An
example is the distance formula d = rt.
When solving these formulas, the idea is to treat all the variables except for the one
that you are solving for as constants. Get the variable that you are solving for on a side
by itself.
For example, solve the standard form of a line 2x + 3y = 21 for y.
2x – 2x + 3y = 21 – 2x
3y = 21 – 2x
3y/3 = 21/3 – 2x/3
y = 7 – 2x/3
side.
y = –2x/3 + 7
or y = –2/3x + 7
First subtract 2x from both sides.
Next divide both sides by 3.
Usually the x term is written first, so rearrange the right
Writing and Graphing Inequalities
A linear inequality is like a linear equation with one difference: an inequality sign (<,
>, <, >,
) replaces the equal sign (=). As an example, y < 2x – 2 is a linear inequality.
Use the methods described for linear equations to plot an inequality. There are,
however, some differences:
•
•
•
Where the (<) or (>) signs are used, test regions above and below or to the left
and right of the line to determine where the inequality is satisfied (e.g., the
solutions). Shade the included regions accordingly. Notice that the line itself is
not part of the solution. Thus, the line should be drawn as a dotted line.
If the (>) or (<) signs are used, proceed as above. In these cases, the line is part
of the solution. Thus, the line should be drawn as a solid line.
If the (
) sign is used, everything except the line is part of the solution to the linear
inequality.
Like linear plots, the plots of linear inequalities are not expected to be symmetric.
The following is a sample plot of a linear inequality:
Solving Inequalities
While an equation states a balanced relationship between expressions, an inequality
states an unequal relationship between two expressions. The following signs are used
to state the relationships that can exist: (<) less than, (<) less than or equal, (>) greater
than, and (>) greater than or equal. Inequalities can be solved in a similar fashion to
how equations are solved. Similar to the two rules for solving equations, there are three
rules for solving inequality.
The first rule states that the same real number can be added to both sides of the
inequality. The second rule states that the same real positive number can be multiplied
by both sides of the inequality. The third rule states that the same real negative number
can be multiplied by both sides of the inequality, but when this multiplication takes
place, the inequality must be reversed. Try some examples.
In the first example, x plus 1 is less than or equal to 2.
x + 1 < 2.
First, demonstrate the addition rule and add negative 1 to both sides.
x + 1 + (–1) < 2 + (–1)
This results in the solution of x is less than or equal to 1.
x<1
In the second example, 5 minus 3x is greater than or equal to 18.
5 – 3x > 18
First, add negative five to both sides.
5 + (–5) – 3x >18 + (–5)
The result is –3x greater than or equal to 13.
–3x > 13
Next, multiply both sides by negative 1 over 3.
(–1/3)(–3x) > 13(–1/3)
x < –13/3