ACT Math - How to simplify an expression

Which of the following expressions is equivalent to 4x2 + 10x – 6?

2(2x – 1)(x + 3)

2(2x +1)(x – 3)

(4x + 2)(x – 3)

(2x – 2)(2x + 3)

(2x +1)(2x – 6)

Explanation

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).

What is the simplified version of the expression:
?

Explanation

Use PEMDAS to dictate which operation comes first. Simplify the parentheses:
and

Next come exponents:

After that comes multiplication and division left to right:

and

Finally, add all the terms together:

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?

250,000

150,000

100,000

50,000

25,000

Explanation

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

Find , given that

$\frac{-2}{7} < x < \frac{10}{7}$

$\frac{-2}{7} < x < \frac{-10}{7}$

$x>\frac{-2}{7}$

$x>\frac{2}{7}$

None of the other answers

Explanation

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.)

Which of the following values is the greatest?

164

324

Explanation

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.

The expression

$\frac{x^{1/3} \cdot 1000y^3 }{(x^3 \cdot y \cdot10y)}$

can be rewritten as:

$\frac{100y}{x^{\frac{8}{3}}}$

$100(x^{\frac{1}{9}})y$

Explanation

To simplify this problem, let’s look at each term individually. $\frac{1000}{10} = 100$ ; $\frac{y^3 }{y\cdot y} = y$ ; $\frac{x^{1/3}}{x^3} =\frac{x^{\frac{1}{3}}}{x^{\frac{9}{3}}} =\frac{1}{x^{\frac{8}{3}}}$ . Thus B is the correct answer.

What is the value of x when 2x - 7 = -5x + 28

-5

-7

Explanation

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.

What is $\small \frac{2(x^{3}y^{4})+2(x^{2}y^{2})}{(4x^{2})(4y^{2})}$ in simplified form?

$\small \frac{xy^{2}+1}{8}$

$\small \frac{2xy^{2}+2}{16}$

$\small \frac{x^{3}y^{4}+x^{2}y^{2}}{4x^{2}y^{2}}$

$\small \frac{xy^{2}}{4}$

$\small \frac{x^{2}y^{2}+1}{4}$

Explanation

Reduce by first dividing all terms by 2: $\small \frac{(x^{3}y^{4})+(x^{2}y^{2})}{8x^{2}y^{2}}$

Next divide the terms in the numerator by the two variables in the denominator. Remember that dividing variables with exponents is really just subtraction of those exponents: $\small \frac{xy^{2}+1}{8}$