substitute our variables into the
equation for the perimeter.
The
formula already calls for
w
, so we
can just leave that as is.
Where it
calls for length, we can just plug
in “2
w
+ 3”
.
We already know
our perimeter is 48ft, so we
substitute that in for
p
.
Use the distributive property to multiply 2 by 2
w
and 3.
Combine like terms (4
w
and 2
w
).
Subtract 6 from both sides to “undo” the addition.
Divide by 6 on both sides to “undo” t
he multiplication.
You are left with a width of 7 ft.
Now we need to find
the length.
w
Check to see that it works
2(7) + 3
14 + 3
17ft = length
Your turn:
I have a box.
The length of the box is 12 in.
The height of the box is 5
in.
The box has a total volume of 360 in.
2
What is the width of the box?
Note:
The formula for volume is V = lwh
,
where v = volume, l =
length, w = width, and h = height.
(You should get
w
= 6in.)
Inequalities
We’ve all
been taught little tricks to remember the inequality sign.
For
example; when given x < 10, we know that x is
less than
ten because x has
the LITTLE side of the sign and 10 has the BIG side of the sign.
Solving inequalities is similar to solving equations; what you do to one side
of an inequality, we must do to the other.
If we are given
x + 7 > 13
and
asked to solve, we would undo the addition on the left side by subtracting 7
from both sides.
We would then be left with
x > 6
, which is our answer.
Suppose we were given
¼ x < 2.
To undo the division, we would multiply
both sides by 4.
The result would be
x < 8
.
But if we had
–
¼ x < 2
, we would multiply by both sided of the inequality
by 4 and the rule is that
when multiplying (or dividing) by a
negative
number
, we must always
flip the sign of an inequality
.
So we would get
x > 2.
Substitute our newly found width and simplify using
order of operations
So now we know that the yard is 7ft wide and 17ft long.
Let’s practice:
It’s always good to check our answer.
To do this, plug values in for x.
Let’s
try
1
, since it is greater than 0 and an easier number to work with.
1
–
5
is 
4.
4 times 2
is 8.
Is
8 < 10?
You bet!
Your turn: 4x + 2 > 10
[You should get x > 2]
Absolute Values
The absolute value of a number is its
distance
from zero on the number line.
Since we
can’t have negative distance, absolute values are always positive.
│5│= 5
│

7│= 7
│

8 + 2│= 6
When solving absolute value equations, we must consider that the number
inside
could have been negative
before you applied the absolute value!
For example:
│x + 2│= 3
x + 2
and
x + 2
3
3
When we take the absolute
value, both are equal to 3.
So we end up with TWO
SOLUTIONS for this
equation.
Solve:
2(x
–
5)
< 10
2
2
x
–
5 > 5
+ 5
+ 5
x > 0
Just as with an equation, we start here by
dividing both sides by 2.
Since we are
dividing by a negative, we must flip the
inequality sign.
Next, we add five to both sides and we
are left with x > 0.
x + 2 = 3
and
x + 2 = 3
1
1
2
2
x = 1
x = 5
Your turn:
│3x –
4│= 5
[You should get x = 1/3 and x = 3]
Linear Equations
If we plot all the solutions to a linear equation on a graph, they form a line.
That is why they are called
line
ar equations
.
Linear equations can be
written in
slopeintercept form
.
There are two important things to know
when writing the equation of a line in
slopeintercept form