Equations / Inequalities - ACT Math

Card 0 of 2799

Question

Find the solution to the following equation if x = 3:

y = (4x2 - 2)/(9 - x2)

Answer

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

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Question

I. x = 0

II. x = –1

III. x = 1

Answer

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Question

Answer

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Question

Given the following system, find the solution:

x = y – 2

2x – 2y = 2

Answer

When 2 equations in a system have the same slopes, they will either have no solution or infinite solutions. Since the y-intercepts are not the same, there is no solution to this system.

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Question

$\small h\left ( x\right )=\frac{28}{x+4}$

$\small \textup{For which of the following values of},x,\textup{is the above function undefined?}$

Answer

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

$\small x+4=0$

$\small x=-4$

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Question

$\frac{x+2}{3}=\frac{x}{3}$ Solve for .

Answer

Cross multiplying leaves , which is not possible.

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Question

Solve:

Answer

First, distribute, making sure to watch for negatives.

Combine like terms.

Subtract 7x from both sides.

Add 18 on both sides and be careful adding integers.

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Question

Solve:

Answer

First, distribute the to the terms inside the parentheses.

Add 6x to both sides.

This is false for any value of . Thus, there is no solution.

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Question

Solve for :

Answer

Like other "solve for x" problems, to begin it, the goal is to get x by itself on one side of the equals sign. In this problem, before doing so, the imaginary -1 in front of (-27x+27) must be distributed.

Once this is done, you may start to try to get x by itself.

${\color{Red} 27x}-32={\color{Red} 27x}-27$

However, when subtracting 27x from either side and doing the same on the other, the 27x term cancels out. As a result, the equation becomes:

We know this is an untrue statement because these numbers are 5 spaces away from each other on the number line. The final answer is No Solution.

$-32\neq-27$

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Question

Given the following system, find the solution:

Answer

When two equations have the same slope, they will have either no solution or infinite solutions. By putting both equations into the form , we get:

and

$y = 2x +\frac{5}{3}$

With the equations in this form, we can see that they have the same slope, but different y-intercepts. Therefore, there is no solution to this system.

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Question

Solve the following equation for :

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

Answer

In order to solve for , we must get by itself on one side of the equation.

$2\left ( x - 4\right ) = \frac{4x + 5}{2}$

First, we can distribute the on the left side of the equal sign and the $\frac{1}{2}$ on the right side.

$2x - 8= 2x +\frac{ 5}{2}$

When we try to get by itself, the terms on each side of the equation cancel out, leaving us with:

$- 8= \frac{ 5}{2}$

We know this is an untrue statement, so there is no solution to this equation.

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Question

Solve $\left | 3-4x\right |<0$ .

Answer

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

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Question

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Answer

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Question

Answer

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Question

John, Jill, and Jack are splitting a pizza. John eats of the pizza and Jill eats of the pizza. How much of the pizza is left for Jack?

Answer

We can write an equation for the amount of pizza eaten, with as the amount left for Jack.

$\frac{1}{2} + \frac{1}{3} + x = 1$

To solve this equation, we must find the lowest common denominator of and . We can list the multiples of and to find the least common multiple:

We can see that the least common multiple of and is , so we can rewrite each of the fractions with a denominator of .

$\frac{1}{2}\cdot \frac{3}{3} = \frac{3}{6}$

$\frac{1}{3}\cdot \frac{2}{2} = \frac{2}{6}$

When we put these fractions back into the equation, we can solve for :

$\frac{3}{6} + \frac{2}{6} + x = 1$

$\frac{5}{6} + x = 1$

$x = \frac{1}{6}$

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Question

What is the slope of a line represented by the equation:

Answer

To solve problems where you need to find the slope of a line in a given equation, change the equation so that it matches y-intercept form:

For this equation, first move the 3x over to the other side of the equation.

The equation should now look like this:

Then, divide by 15 to isolate the variable .

$\frac{15y}{15} =-\frac{3x}{15} + \frac{4}{15} \rightarrow y =-\frac{3}{15}x + \frac{4}{15}$

Then simplify

$y =-\frac{3}{15}x + \frac{4}{15} \rightarrow y =-\frac{1}{5}x + \frac{4}{15}$

Whatever number is before the x in the equation (m) is your slope.

$-\frac{1}{5}$

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Question

If 6_x_ = 42 and xk = 2, what is the value of k?

Answer

Solve the first equation for x by dividing both sides of the equation by 6; the result is 7. Solve the second equation for k by dividing both sides of the equation by x, which we now know is 7. The result is 2/7.

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Question

If 4_x_ + 5 = 13_x_ + 4 – x – 9, then x = ?

Answer

Start by combining like terms.

4_x_ + 5 = 13_x_ + 4 – x – 9

4_x_ + 5 = 12_x_ – 5

–8_x_ = –10

x = 5/4

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Question

If 3 – 3_x_ < 20, which of the following could not be a value of x?

Answer

First we solve for x.

Subtracting 3 from both sides gives us –3_x_ < 17.

Dividing by –3 gives us x > –17/3.

–6 is less than –17/3.

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