Simplifying Expressions - ACT Math
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If a = 4 and b =3 then: 2a2 + 3ab – 7 is?
If a = 4 and b =3 then: 2a2 + 3ab – 7 is?
Substitute the values of a and b into the equation. Then
2a2 + 3ab – 7
(2 x 42 ) + (3 x 4 x 3) – 7
(2 x 16) + (36) -7
32 + 36 – 7
61
Substitute the values of a and b into the equation. Then
2a2 + 3ab – 7
(2 x 42 ) + (3 x 4 x 3) – 7
(2 x 16) + (36) -7
32 + 36 – 7
61
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Which of the following is equivalent to 
?
Which of the following is equivalent to ?
First, we can use the property of exponents that xy/xz = xy–z
Then we can use the property of exponents that states x–y = 1/xy
a–1b5c–1 = b5/ac
First, we can use the property of exponents that xy/xz = xy–z
Then we can use the property of exponents that states x–y = 1/xy
a–1b5c–1 = b5/ac
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What is the value of x when 3x + 5 = 2x – 7?
What is the value of x when 3x + 5 = 2x – 7?
To answer this question we need to isolate x. A useful first step is to subtract 5 from both sides. The expression then becomes 3x = 2x – 12. Then we can subtract 2x from both sides. This leaves x = –12.
To answer this question we need to isolate x. A useful first step is to subtract 5 from both sides. The expression then becomes 3x = 2x – 12. Then we can subtract 2x from both sides. This leaves x = –12.
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Simplify the following expression:
2x(x2 + 4ax – 3a2) – 4a2(4x + 3a)
Simplify the following expression:
2x(x2 + 4ax – 3a2) – 4a2(4x + 3a)
Begin by distributing each part:
2x(x2 + 4ax – 3a2) = 2x * x2 + 2x * 4ax – 2x * 3a2 = 2x3 + 8ax2 – 6a2x
The second:
–4a2(4x + 3a) = –16a2x – 12a3
Now, combine these:
2x3 + 8ax2 – 6a2x – 16a2x – 12a3
The only common terms are those with a2x; therefore, this reduces to
2x3 + 8ax2 – 22a2x – 12a3
This is the same as the given answer:
–12a3 – 22a2x + 8ax2 + 2x3
Begin by distributing each part:
2x(x2 + 4ax – 3a2) = 2x * x2 + 2x * 4ax – 2x * 3a2 = 2x3 + 8ax2 – 6a2x
The second:
–4a2(4x + 3a) = –16a2x – 12a3
Now, combine these:
2x3 + 8ax2 – 6a2x – 16a2x – 12a3
The only common terms are those with a2x; therefore, this reduces to
2x3 + 8ax2 – 22a2x – 12a3
This is the same as the given answer:
–12a3 – 22a2x + 8ax2 + 2x3
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What is the value of x when 2x - 7 = -5x + 28
What is the value of x when 2x - 7 = -5x + 28
We need to solve for x. To do so, first move the x values to the left of the = sign and all other values to the right. 2x + 5x = 28 + 7. Then add like terms: 7x = 35. In order to get x all by itself, we then divide both sides by 7. X = 5.
We need to solve for x. To do so, first move the x values to the left of the = sign and all other values to the right. 2x + 5x = 28 + 7. Then add like terms: 7x = 35. In order to get x all by itself, we then divide both sides by 7. X = 5.
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If 4x – 6 = 18, what is 3x3 – 4x + 1?
If 4x – 6 = 18, what is 3x3 – 4x + 1?
First we have to solve for x using the first equation:
4x-6 = 18
4x = 24
x = 6
Next, we have to plug in our value for x into the second equation:
3(6)3 – 4(6) + 1
3 (216) – 24 +1 = 625
First we have to solve for x using the first equation:
4x-6 = 18
4x = 24
x = 6
Next, we have to plug in our value for x into the second equation:
3(6)3 – 4(6) + 1
3 (216) – 24 +1 = 625
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What is the value of z when 2z + 3 = 3z – 5?
What is the value of z when 2z + 3 = 3z – 5?
Solve one equation with one variable by moving the constants to one side of the equal sign and the variables to the other. 2z + 3 = 3z – 5 is solved by adding 5 -2z to both side of the equation and ending up with z = 8
Solve one equation with one variable by moving the constants to one side of the equal sign and the variables to the other. 2z + 3 = 3z – 5 is solved by adding 5 -2z to both side of the equation and ending up with z = 8
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Find 
, given that 
Find
Create two equations to eliminate the absolute value function, one where the value inside the absolute value bars is assumed to be positive and another where it is assumed to be negative: 7x – 4 + 5 > –1 and -7x + 4 + 5 > –1.
The solutions for the equations, respectively, are x > -2/7 and x < -10/7. (Remember to flip the inequality sign when multiplying or dividing by a negative number.)
Create two equations to eliminate the absolute value function, one where the value inside the absolute value bars is assumed to be positive and another where it is assumed to be negative: 7x – 4 + 5 > –1 and -7x + 4 + 5 > –1.
The solutions for the equations, respectively, are x > -2/7 and x < -10/7. (Remember to flip the inequality sign when multiplying or dividing by a negative number.)
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Given that 3_x_ + 7 = 4_x_ – 2, what is the value of x ?
Given that 3_x_ + 7 = 4_x_ – 2, what is the value of x ?
First, we must solve the equation for x by subtracting 3_x_ from both sides:
3_x_ – 3_x_ + 7 = 4_x_ – 3_x_ – 2
7 = x – 2
Then we must add 2 to both sides:
7 + 2 = x – 2 + 2
9 = x
First, we must solve the equation for x by subtracting 3_x_ from both sides:
3_x_ – 3_x_ + 7 = 4_x_ – 3_x_ – 2
7 = x – 2
Then we must add 2 to both sides:
7 + 2 = x – 2 + 2
9 = x
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Simplify the following expression: x3 - 4(x2 + 3) + 15
Simplify the following expression: x3 - 4(x2 + 3) + 15
To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x2 and -4 by 3.
x3 - 4x2 -12 + 15
You can then add -12 and 15, which equals 3.
You now have x3 - 4x2 + 3 and are finished. Just a reminder that x3 and 4x2 are not like terms as the x’s have different exponents.
To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x2 and -4 by 3.
x3 - 4x2 -12 + 15
You can then add -12 and 15, which equals 3.
You now have x3 - 4x2 + 3 and are finished. Just a reminder that x3 and 4x2 are not like terms as the x’s have different exponents.
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Which of the following does not simplify to 
?
Which of the following does not simplify to
5x – (6x – 2x) = 5x – (4x) = x
(x – 1)(x + 2) - x2 + 2 = x2 + x – 2 – x2 + 2 = x
x(4x)/(4x) = x
(3 – 3)x = 0x = 0
5x – (6x – 2x) = 5x – (4x) = x
(x – 1)(x + 2) - x2 + 2 = x2 + x – 2 – x2 + 2 = x
x(4x)/(4x) = x
(3 – 3)x = 0x = 0
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Which of the following expressions are equivalent to b for all non-zero real numbers a, b, x, and y such that 
Which of the following expressions are equivalent to b for all non-zero real numbers a, b, x, and y such that
Cross multiply to get 2xb = 3ya2, then divide by 2x to get b on one side of the equal sign by itself.
Cross multiply to get 2xb = 3ya2, then divide by 2x to get b on one side of the equal sign by itself.
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A store in California specializes in the sale of custom t-shirts and is growing rapidly. In their second year of business, they doubled the sales from the first year, and in the third year, they sold 50,000 more t -shirts than their second year. In their fourth year, they doubled their sales from the third year. If in the fourth year of business, the store sold 300,000 shirts, how many did they sell in the second year?
A store in California specializes in the sale of custom t-shirts and is growing rapidly. In their second year of business, they doubled the sales from the first year, and in the third year, they sold 50,000 more t -shirts than their second year. In their fourth year, they doubled their sales from the third year. If in the fourth year of business, the store sold 300,000 shirts, how many did they sell in the second year?
This question is a bit wordy, so it is tough to not get lost in it, it usually helps to write down the pertinent information on a separate sheet of paper. In this case, you should have written that year 4 = 2 x year 3 = 300,000
so, 300,000 = 2 x year 3
divide each side by 2 and we get
150,000 sold in year 3. We are told that year 3 = year 2 +50,000
so, subtracting 50,000 from each side yields
year 3 - 50,000 = total sales from year 2
150,000 - 50,000 = total sales for year 2 = 100,000
This question is a bit wordy, so it is tough to not get lost in it, it usually helps to write down the pertinent information on a separate sheet of paper. In this case, you should have written that year 4 = 2 x year 3 = 300,000
so, 300,000 = 2 x year 3
divide each side by 2 and we get
150,000 sold in year 3. We are told that year 3 = year 2 +50,000
so, subtracting 50,000 from each side yields
year 3 - 50,000 = total sales from year 2
150,000 - 50,000 = total sales for year 2 = 100,000
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a # b = (a * b) + a
What is 3 # (4 # 1)?
a # b = (a * b) + a
What is 3 # (4 # 1)?
Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:
4 # 1 = (4 * 1) + 4 = 4 + 4 = 8
That means: 3 # (4 # 1) = 3 # 8. Solve this now:
3 # 8 = (3 * 8) + 3 = 24 + 3 = 27
Work from the "inside" outward. Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:
4 # 1 = (4 * 1) + 4 = 4 + 4 = 8
That means: 3 # (4 # 1) = 3 # 8. Solve this now:
3 # 8 = (3 * 8) + 3 = 24 + 3 = 27
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Which of the following values is the greatest?
Which of the following values is the greatest?
In order to compare each quantity, we need to rewrite each one using the same base. A common base that would be easy to use would be 2, because we can write 2, 4, 8, 16, and 32 all as a power of 2. We also need to remember that when taking an exponent to an exponent, we have to multiple the two exponents together.
28 = 28
82 = (23)2 = 26
164 = (24)4 = 216
49 = (22)9 = 218
324 = (25)4 = 220
When we compare all of the numbers as powers of 2, we realize that 220 is the largest. Thus, the answer is 324.
In order to compare each quantity, we need to rewrite each one using the same base. A common base that would be easy to use would be 2, because we can write 2, 4, 8, 16, and 32 all as a power of 2. We also need to remember that when taking an exponent to an exponent, we have to multiple the two exponents together.
28 = 28
82 = (23)2 = 26
164 = (24)4 = 216
49 = (22)9 = 218
324 = (25)4 = 220
When we compare all of the numbers as powers of 2, we realize that 220 is the largest. Thus, the answer is 324.
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Which of the following is equivalent to (x)(x)(x)(x)(x**–**2)?
Which of the following is equivalent to (x)(x)(x)(x)(x**–**2)?
When multiplying powers of x, we add the exponents. The first four terms are equivalent to x4.
When multiplying powers of x, we add the exponents. The first four terms are equivalent to x4.
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Which of the following expressions is equivalent to 4x2 + 10x – 6?
Which of the following expressions is equivalent to 4x2 + 10x – 6?
First, pull out a common factor of 2 to get 2(2x2 + 5x – 3). Then factor the quadratic so that the x terms add to 5x and the numbers multiply to - 3, resulting in 2(2x – 1)(x + 3).
First, pull out a common factor of 2 to get 2(2x2 + 5x – 3). Then factor the quadratic so that the x terms add to 5x and the numbers multiply to - 3, resulting in 2(2x – 1)(x + 3).
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If 
and 
, then 
If
We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.
Notice what happens if we line up the two equations and add them together.
(x + y) + (3_x –_ y + z) = 4x + z
and 5 + 3 = 8
Lets take this equation and multiply the whole thing by 3:
3(4_x_ + z = 8)
Thus, 12_x_ + 3_z_ = 24.
We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.
Notice what happens if we line up the two equations and add them together.
(x + y) + (3_x –_ y + z) = 4x + z
and 5 + 3 = 8
Lets take this equation and multiply the whole thing by 3:
3(4_x_ + z = 8)
Thus, 12_x_ + 3_z_ = 24.
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Simplify the result of the following steps, to be completed in order:
1. Add 7_x_ to 3_y_
2. Multiply the sum by 4
3. Add x to the product
4. Subtract x – y from the sum
Simplify the result of the following steps, to be completed in order:
1. Add 7_x_ to 3_y_
2. Multiply the sum by 4
3. Add x to the product
4. Subtract x – y from the sum
Step 1: 7_x_ + 3_y_
Step 2: 4 * (7_x_ + 3_y_) = 28_x_ + 12_y_
Step 3: 28_x_ + 12_y_ + x = 29_x_ + 12_y_
Step 4: 29_x_ + 12_y_ – (x – y) = 29_x_ + 12_y_ – x + y = 28_x_ + 13_y_
Step 1: 7_x_ + 3_y_
Step 2: 4 * (7_x_ + 3_y_) = 28_x_ + 12_y_
Step 3: 28_x_ + 12_y_ + x = 29_x_ + 12_y_
Step 4: 29_x_ + 12_y_ – (x – y) = 29_x_ + 12_y_ – x + y = 28_x_ + 13_y_
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Suzanne is at the grocery store. She has $5.00 to spend on produce. Oranges are $2.50 per pound, apples cost $1.50 per pound and bananas are $0.50 per pound. Which combination of fruit will fit her budget?
Suzanne is at the grocery store. She has $5.00 to spend on produce. Oranges are $2.50 per pound, apples cost $1.50 per pound and bananas are $0.50 per pound. Which combination of fruit will fit her budget?
Make a simple algebra equation and test it against each combination:
Total Cost = $2.50 * (# Oranges) + $1.50 * (# Apples) + $0.50 * (# Bananas)
Make a simple algebra equation and test it against each combination:
Total Cost = $2.50 * (# Oranges) + $1.50 * (# Apples) + $0.50 * (# Bananas)
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