Transcript PC4.7

4.7
Radical Equations and
Problem Solving
Power Rule
• When solving radical equations, we
use a new principle called the power
rule.
– The Power Rule states that if a = b, then
an = bn and vice versa.
• This also applies to radicals,
specifically:
n
n
a  b.
– If a = b, then
Example 1
• If a radical with a variable is already
isolated in an equation, we will raise each
side of the equation to a power that
matches the radical index.
• Solve the radical equations below.
x5 6

3

j 2  3
Extraneous Solutions
• As was the case with rational equations, it was
necessary to check answers to be sure they
verified the original equation.
• With radical equations, it is also necessary for
two primary reasons:
– With an even index, we cannot have a negative
radicand (i.e. we cannot take the square root of a
negative)
– Unless specified otherwise, we will only calculate the
principal root of any radical.
• Example 2: Solve
solutions.
x  3  7and check your
Solving polynomial inequalities
• Rewrite the polynomial so that all terms are
on one side and zero on the other.
• Factor the polynomial. We are interested in
when factors are either pos. or neg., so we
must know when the factor equals zero.
• The values of x for which the factors equal
zero are the boundary points, which we place
on the number line.
• The intervals around the boundary points
must be tested to find on which interval(s)
will the polynomial be positive/negative.
Solve : (x – 3)(x + 1)(x – 6) < 0
• To solve this inequality we observe that 0 is
already on one side and the polynomial is
factored already on the other side.
• The 3 boundary values are x = 3,-1,6
• They create 4 intervals:
(,1), (1,3), (3,6), (6, )
• Pick a number in each interval to test the sign of
that interval. If the polynomial is negative
there then the interval is in the solution set.
• Solution set:
Solve: x3 +3x2 ≥ 10x
1. To solve, first we must rewrite the
inequality so all terms are on one side
and 0 on the other, then factor.
2. x(x-2)(x+5) ≥ 0
3. Boundary points: 0, 2, -5
4. Solution set:
Solving rational inequalities
• VERY similar to solving polynomial inequalites
EXCEPT if the denominator equals zero,
there is a domain restriction. The function
COULD change signs on either side of that
point.
• Step 1: Rewrite the inequality so all terms
are on one side and zero on the other.
• Step 2: Factor both numerator &
denominator to find boundary values for
regions to check when function becomes
positive or negative. And do as before !
Solve the following inequalities:
1)
2)
3)
x 1
0
3  4x
2x
0
2
9 x
3
7

x x 1