What is the rule for solving the absolute value equation |x| = a?
- x = a
- x = -a
This rule states that the variable can equal either the positive or negative value of ‘a’.
Solve the absolute value equation |x + 4| = 9.
- x + 4 = 9
- x + 4 = -9
- x = 5
- x = -13
The solutions are derived by setting the expression inside the absolute value equal to both 9 and -9.
What is the solution set for the inequality |x| ≤ a?
-a ≤ x ≤ a
This indicates that x is bounded between -a and a, inclusive.
What is the solution set for the inequality |x| ≥ a?
- x ≤ -a
- x ≥ a
This means x is either less than or equal to -a or greater than or equal to a.
Solve the inequality |x + 4| ≤ 9.
-9 ≤ x + 4 ≤ 9
* -13 ≤ x ≤ 5
The solution is found by breaking the compound inequality into two parts.
Solve the inequality |x + 4| ≥ 9.
- x + 4 ≤ -9
- x ≤ -13
- x + 4 ≥ 9
- x ≥ 5
- (-∞, -13] U [5, ∞)
The solution includes two intervals where x is either less than or equal to -13 or greater than or equal to 5.
What is a monomial in algebra?
An expression that contains one term, including numbers, variables, or a combination of them
Examples include 3, x, 98b, mn, Sxy, or 3x³y⁵.
What is the degree of a monomial?
The sum of the exponents of the variables in the monomial
The degree for a constant is always 0, and for a variable without an exponent listed, it is always 1.
In the degrees of a monomial, the degree of a constant is always ?
Zero
Example:
3a^2b^5c
The degree of number three is zero because it is a constant
In monomials the degree for a variable that doesn’t have an exponent is always ?
One
Example:
3a^2b^5c
The degree of the variable c is 1
Find the degree of the following monomial
:
3a^2b^5c
8
Evaluate: 1^{25}
1
Any number raised to any power equals 1.
Evaluate: 7^0
1
Any non-zero number raised to the power of 0 equals 1.
Evaluate: 12^1
12
Any number raised to the power of 1 equals the number itself.
Simplify: x^4 multiplied by x^7
x^{11}
When multiplying like bases, add the exponents.
Simplify: x^9 divided by x^5
x^{4}
When dividing like bases, subtract the exponents.
Simplify: (x^3)^4
x^{12}
When raising a power to a power, multiply the exponents.
Write in fraction form: ** {x^5}over{y^5}**
(x/y)^5
The quotient of like bases can be expressed as a single base raised to the exponent.
Evaluate: (-3)^5
-243 (negative)
A negative number raised to an odd power remains negative.
Evaluate: (-4)^4
256 (positive)
A negative number raised to an even power becomes positive.
Rewrite using positive exponent: x^{-6}
1/x^6
A negative exponent indicates the reciprocal of the base raised to the positive exponent.
Rewrite: {5}over{9}
^{3}
{9}{5}
^{-3}
A negative exponent inverts the fraction and changes the exponent to positive.
If x^a = x^b, then __________.
a = b
This property holds true for any non-zero base.
