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 Equations/149693: This question is from textbook Thank you in advance for helping me with writing this problem in interval notation. The absolute value of 5x-3=101 solutions Answer 109817 by jim_thompson5910(28476)   on 2008-07-25 00:56:55 (Show Source): You can put this solution on YOUR website! Start with the given equation Break up the absolute value (remember, if you have , then or ) or Set the expression equal to the original value 10 and it's opposite -10 Now lets focus on the first equation Add 3 to both sides Combine like terms on the right side Divide both sides by 5 to isolate x Reduce Now lets focus on the second equation Add 3 to both sides Combine like terms on the right side Divide both sides by 5 to isolate x Reduce So the solutions to are: and
 Inequalities/149696: How do you write an inequality using x Example:i have at least $25 in my pocket 1 solutions Answer 109816 by jim_thompson5910(28476) on 2008-07-25 00:55:05 (Show Source): You can put this solution on YOUR website!Let x=amount of money in your pocket. So when you have "at least$25", this means that you have 25 or more. Mathematically speaking, this means that some unknown quantity "x" is greater than or equal to 25. So symbolically, this inequality is
 Inequalities/149695: This question is from textbook Thank you in advance for helping me with writing this problem in interval notation. The absolute value of 5n-2<2 I have 01 solutions Answer 109815 by jim_thompson5910(28476)   on 2008-07-25 00:52:33 (Show Source): You can put this solution on YOUR website! Start with the given inequality Break up the absolute value (remember, if you have , then and ) and Break up the absolute value inequality using the given rule Combine the two inequalities to get a compound inequality Add 2 to all sides Divide all sides by 5 to isolate n ---------------------------------------------------- Answer: So our answer is which looks like this in interval notation
 Geometric_formulas/149702: The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into two segments, whose lengths are 8" and 18". How long is the altitude? 1 solutions Answer 109813 by jim_thompson5910(28476)   on 2008-07-25 00:44:23 (Show Source): You can put this solution on YOUR website!First, let's draw the picture. Be sure you label every length you can So we can see that the hypotenuse of the largest triangle is 26 units. Using pythagoreans theorem, we get: Now looking at the left most smaller triangle, we see that the legs are 8 and "h" while the hypotenuse is "x". So once again, with pythagoreans theorem, we can say: Now solving for , we get Finally, the right most smaller triangle has a base of 18 and "h" and a hypotenuse of y. So this gives us: --------- Start with the first equation. Plug in . In other words, replace with Square 18 to get 324. Square 26 to get 676. Square 18 to get 324. Square 26 to get 676. Plug in Square 8 to get 64 Combine like terms. Subtract 260 from both sides. Divide both sides by 2. Take the square root of both sides. Note: only the positive square root is considered. So the length of x is which is about 14.422 units (this isn't important). Go back to the second equation Plug in Square the square root to eliminate it. Square 8 to get 64 Combine like terms. Take the square root of both sides. Once again, only the positive square root is considered. So the height is 12 units which means that altitude is 12 units.
 Geometric_formulas/149701: The dimensions of rectangle ABCD are AB=12 and BC=16. Point P is marked on side BC so that BP=5, and the intersection of AP and BD is called T. Find the lengths of the four segments TA, TP, TB, and TD. 1 solutions Answer 109812 by jim_thompson5910(28476)   on 2008-07-25 00:42:00 (Show Source): You can put this solution on YOUR website!First draw the rectangle with the point P Now draw in the segments AP, BD, and PD. Take note of the unknown variables I'm assigning. If you look closely, you'll notice that a trapezoid forms due to these extra lines segments. Through the use of pythagoreans theorem, we get the length of AP of 13 and BD of 20 Now, it turns out that the ratio of the parallel sides 5 and 16 are the same as the ratio of the lengths of the cut diagonals. So the following ratios are true: and So let's solve for x: Start with the first ratio Cross multiply Distribute and multiply. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. So the approximate length of x is 3.095 units. This means that TP is 3.095 units. This means that the other length is . So TA is 9.905 units. ------------------ Now let's solve for y: Start with the second ratio Cross multiply Distribute and multiply. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. So the approximate length of y is 4.762 which means that the other length is . So TB is 4.762 units and TD is 15.238 units.
 Geometric_formulas/149700: The parallel sides of a trapezoid are 12" and 18" long. The non-parallel sides meet when one is extended 9" and the other is extended 16". How long are the non-parallel sides of this trapezoid.1 solutions Answer 109811 by jim_thompson5910(28476)   on 2008-07-25 00:38:52 (Show Source): You can put this solution on YOUR website!If we draw the picture, we get From the picture, we can see that there is one large triangle and there is one smaller triangle towards the top. It turns out that these triangles are similar. So this means that these ratios hold: and So let's find the length of x: Start with the first ratio. Cross multiply Distribute and multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Divide. So the length of x is 4.5 units. ----------------- Now let's find the length of y: Start with the second ratio Cross multiply Distribute and multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So the length of y is 8 units. So the lengths of the non-parallel sides are 4.5 and 8 units.
 Geometric_formulas/149699: The vectors (8,0) and (3,4) form a paralellogram. Find the lengths of its altitudes. I don't know how to generate the algebraic equations to solve this one.1 solutions Answer 109810 by jim_thompson5910(28476)   on 2008-07-25 00:33:06 (Show Source): You can put this solution on YOUR website!First plot the two points (0,0) and (3,4). Now draw a right triangle with the hypotenuse that goes through the two points. With the use of pythagoreans theorem, we find that the hypotenuse is 5 units (since ). So our drawing looks like this: Now if draw the vectors and draw the parallelogram, we can see that the height of the parallelogram is the y-coordinate of the point (3,4). So the first height is 4. So from the drawing, we see that the base is 8 units and the height is 4 units. Now if we compute the area, we get: Now let the vector from (0,0) to (3,4) be the new base. So the base is now 5. The area is still 32. So this means that Solving for h, we get So the second height is 6.4
 Equations/149663: Solve for x 6x+1=6x-81 solutions Answer 109797 by jim_thompson5910(28476)   on 2008-07-24 18:17:36 (Show Source): You can put this solution on YOUR website! Start with the given equation. Subtract from both sides. Subtract from both sides. Combine like terms on the left side. Combine like terms on the right side. Simplify. Since this equation is never true for any x value, this means that there are no solutions.
 Evaluation_Word_Problems/149664: Solve for x -8x+14=-2(4x-7)1 solutions Answer 109796 by jim_thompson5910(28476)   on 2008-07-24 18:16:37 (Show Source): You can put this solution on YOUR website! Start with the given equation. Distribute. Subtract from both sides. Add to both sides. Combine like terms on the left side. Combine like terms on the right side. Simplify. Since this equation is always true for any x value, this means x can equal any number. So there are an infinite number of solutions.
 Proportions/149638: how do you do this? Find the missing number: 20/32 = ?/241 solutions Answer 109773 by jim_thompson5910(28476)   on 2008-07-24 16:07:27 (Show Source): You can put this solution on YOUR website! Start with the given ratio Multiply both sides by 24 Multiply Simplify So our answer is
 Equations/149635: Add and Simplify: 1/x-2 - 4/x^2-4 + 3/x+21 solutions Answer 109766 by jim_thompson5910(28476)   on 2008-07-24 15:45:18 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor to get So the LCD is Multiply the first fraction by . Multiply the third fraction by . Doing this will make each denominator the LCD Combine the fractions. Distribute Since the denominators are equal, we can combine the numerators of the common denominator. Combine like terms. Factor out the GCF 4 from the numerator. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to In other words, where or
 Equations/149627: Add and simplify: 1/x-2 - 4/x^2 + 3/x+2 Multiply and simplify: x^2-6x+8/3x+9 multiplied by x+3/x^2-4 Divide and simplify: y2+7y +10/2y-4 divided by y^2-3y-10/y-21 solutions Answer 109764 by jim_thompson5910(28476)   on 2008-07-24 15:19:32 (Show Source): You can put this solution on YOUR website!# 1 Start with the given expression. Multiply the first term by . Multiply the second term by . Multiply the third term by . Combine the fractions. FOIL the terms in the numerator. Distribute. Combine the fractions. Distribute Combine like terms. So simplifies to In other words, where , , or # 2 Start with the given expression. Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where , , or # 3 Start with the given expression. Multiply the first fraction by the reciprocal of the second fraction . Factor to get . Factor to get . Factor to get . Combine the fractions. Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to . In other words, where , , or
 Quadratic_Equations/149630: Find a quadratic equation that has solutions 5 and -61 solutions Answer 109763 by jim_thompson5910(28476)   on 2008-07-24 15:03:08 (Show Source): You can put this solution on YOUR website! Since and are given zeros this means that: and Get all terms to the left side in each case and Now use the zero product property in reverse to join the factors. Expand and multiply ------------------------------------------- Answer: So the polynomial with roots of and is Notice how if we graph , we can visually verify our answer Graph of with roots of and
 Circles/149615: What is the circumference of a circle with a radius of 4.1 m? Round to the nearest tenth.Use pi=3.14 Thanks, Amanda1 solutions Answer 109756 by jim_thompson5910(28476)   on 2008-07-24 14:15:45 (Show Source): You can put this solution on YOUR website! Start with the circumference of a circle formula Plug in Multiply 2, and (you can use 3.14 for pi) to get So the circumference of a circle with a radius of 4.1 meters is 25.8 meters (rounded to the nearest tenth)
 logarithm/149609: How would you solve: log(base 2x)+log(base 4x)+log(base 8x) = 1?1 solutions Answer 109750 by jim_thompson5910(28476)   on 2008-07-24 14:02:18 (Show Source): You can put this solution on YOUR website! Start with the given equation. Use the Change of Base formula to rewrite each log. Remember, the Change of Base formula is: Rewrite as . Rewrite as . Rewrite each log using the identity Now to make things simple, let . So this means that the equation is now Notice now that the LCD is 6z Multiply both sides by the LCD to clear the fractions. Distribute and multiply. Factor out the GCF Add Rearrange the terms. Divide both sides by 11. Now replace "z" with . Rearrange the terms. Divide both sides by . Use the change of base formula to rewrite the left side. Rewrite the equation using the property: ====> So the answer is which approximates to
Quadratic_Equations/149608: What three techniques can be used to solve quadratic equations? Demonstrate these techniques on the equation 12x^2-10x-42=0.
1 solutions

Answer 109749 by jim_thompson5910(28476)   on 2008-07-24 13:49:44 (Show Source):
You can put this solution on YOUR website!
Technique #1 Factoring:

First let's factor

Factor out the GCF

Now let's focus on the inner expression

------------------------------------------------------------

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :
1,2,3,6,7,9,14,18,21,42,63,126
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*(-126)
2*(-63)
3*(-42)
6*(-21)
7*(-18)
9*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First NumberSecond NumberSum
1-1261+(-126)=-125
2-632+(-63)=-61
3-423+(-42)=-39
6-216+(-21)=-15
7-187+(-18)=-11
9-149+(-14)=-5
-1126-1+126=125
-263-2+63=61
-342-3+42=39
-621-6+21=15
-718-7+18=11
-914-9+14=5

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

So factors to

Set the factored expression equal to zero

Now set each factor equal to zero:

or

or Now solve for x in each case

or

Let's use the quadratic formula to solve for x

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

Technique # 3 Completing the square

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Notice how . Since we're adding 0, we're not changing the equation.

Now factor to get

Combine like terms

Distribute

Multiply

So after completing the square, becomes .

So is equivalent to

Divide both sides by 12.

Take the square root of both sides.

or Break up the expression

or Take the square root of to get

or Subtract from both sides.

or Combine like terms and simplify.

Technique # 4 Graphing

Simply graph to get

Graph of

Now use the calculator's zero function to find the zeros at and

 Distributive-associative-commutative-properties/149588: Simplify the expression 3(2-x)-2(3-x)1 solutions Answer 109747 by jim_thompson5910(28476)   on 2008-07-24 13:36:34 (Show Source): You can put this solution on YOUR website! Start with the given expression. Distribute. Group like terms. Combine like terms. Simplify So