would really appreciate help on these equations.
3. Devon tosses a horseshoe at a stake 30 feet away. The horseshoe lands no more than 3 feet from the stake. (a) Write an absolute value inequality that represents the range of distances that the horseshoe travels. (b) Solve the inequality.
Solve for x
4. (x - 2)(x + 1) = 4
Factor completey
7. 2x^4 + 16x
3. (a) To represent the range of distances the horseshoe can travel, we use the absolute value inequality.
Let's denote the distance the horseshoe travels as "d." Since the horseshoe lands no more than 3 feet from the stake, the inequality can be written as:
|d - 30| ≤ 3
This inequality states that the distance "d" minus 30 is less than or equal to 3 in absolute value.
(b) To solve the absolute value inequality, we can split it into two separate inequalities:
1) d - 30 ≤ 3
To solve for "d," we add 30 to both sides:
d ≤ 33
2) -(d - 30) ≤ 3
First, distribute the negative sign:
-d + 30 ≤ 3
Next, subtract 30 from both sides:
-d ≤ -27
Finally, multiply both sides by -1 to switch the inequality sign:
d ≥ 27
Therefore, the solution to the absolute value inequality |d - 30| ≤ 3 is 27 ≤ d ≤ 33.
4. To solve for x in the equation (x - 2)(x + 1) = 4, we can follow these steps:
Expand the equation: x^2 - 2x + x - 2 = 4
Combine like terms: x^2 - x - 2 = 4
Move all the terms to one side of the equation: x^2 - x - 2 - 4 = 0
Simplify: x^2 - x - 6 = 0
Now, we need to factor the quadratic equation:
(x - 3)(x + 2) = 0
Setting each factor equal to zero:
x - 3 = 0 or x + 2 = 0
Solve for x:
x = 3 or x = -2
Therefore, the solutions for the equation (x - 2)(x + 1) = 4 are x = 3 and x = -2.
7. To factor completely the expression 2x^4 + 16x, we can follow these steps:
First, factor out the common factor "2x" from both terms:
2x(x^3 + 8)
Next, we have a sum of cubes in the parentheses. The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Using this formula, we can rewrite the expression as:
2x[(x)^3 + (2)^3]
Factor out the sum of cubes:
2x[(x + 2)(x^2 - 2x + 4)]
Therefore, the completely factored form of 2x^4 + 16x is 2x(x + 2)(x^2 - 2x + 4).
3. (a) To write an absolute value inequality that represents the range of distances that the horseshoe travels, we can consider the horseshoe landing no more than 3 feet from the stake.
Let's assume the distance the horseshoe travels as "d". The absolute value of "d" represents the distance from the stake, regardless of it being a positive or negative value. So the inequality is:
|d| ≤ 3
(b) To solve the inequality, we need to consider two cases: when "d" is positive and when "d" is negative.
Case 1: d is positive
d ≤ 3
Case 2: d is negative
-d ≤ 3
To solve for "d" in both cases, we multiply the second inequality by -1 to flip the inequality sign:
d ≥ -3
Combining both cases, we have:
-3 ≤ d ≤ 3
Therefore, the range of distances that the horseshoe travels is between -3 and 3 feet from the stake.
4. To solve the equation (x - 2)(x + 1) = 4, we can start by applying the distributive property to simplify the equation:
x^2 + x - 2x - 2 = 4
Combine like terms:
x^2 - x - 2 = 4
Now, we can move all terms to one side to get a quadratic equation in standard form:
x^2 - x - 2 - 4 = 0
x^2 - x - 6 = 0
To solve this quadratic equation, we can either factor it completely or use the quadratic formula.
To factor the quadratic equation, we look for two numbers whose sum is -1 (-b coefficient) and whose product is -6 (c coefficient):
(x - 3)(x + 2) = 0
Now, set each factor equal to zero:
x - 3 = 0 or x + 2 = 0
Solving for x:
x = 3 or x = -2
Therefore, the solutions to the equation are x = 3 and x = -2.
7. To factor the expression 2x^4 + 16x, we can start by factoring out the greatest common factor (GCF). The GCF here is 2x, so we can write:
2x(x^3 + 8)
Now, we need to recognize that the expression within the parentheses is a sum of cubes, which can be factored using the formula:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In this case, a = x and b = 2, so we have:
2x(x + 2)(x^2 - 2x + 4)
Therefore, the completely factored form of the expression 2x^4 + 16x is 2x(x + 2)(x^2 - 2x + 4).