ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

varsity tutors app store varsity tutors android store varsity tutors ibooks store

Example Questions

Example Question #18 : Inequalities

Solve:

Possible Answers:

Correct answer:

Explanation:

First, we want to group all of our like terms.  I will move all of my integers to the left side of the inequality. 

Since we are not dividing by a negative sign, we do not have to flip the inequality. 

Example Question #15 : How To Find The Solution To An Inequality With Addition

Solve:

Possible Answers:

Correct answer:

Explanation:

The first thing that we have to do is deal with the absolute value.  We simply remove the absolute value by equating the left side with the positive and negative solution (of the right side). When we include the negative solution, we must flip the direction of the inequality. Shown explicitly:

Now, we simply solve the inequality by moving all of the integers to the right side, and we are left with:   This reduces down to 

Example Question #441 : Algebra

Solve the following inequality:

Possible Answers:

Correct answer:

Explanation:

To solve, simply treat it as an equation. This means you want to isolate the variable on one side and move all other constants to the other side through opposite operation manipulation.

Remember, you only flip the inequality sign if you multiply or divide by a negative number.

Thus,

Example Question #442 : Algebra

Let x be a number such that x > 1 and x < 0. Let y be a number such that y > 0. Which of the following must be true?

Possible Answers:

(xy)2 > 1

x + y > 0

(x/y) < 1

1 < xy < 1

(x + y)2 > 1

Correct answer:

(x/y) < 1

Explanation:

Let us see if we can find counterexamples for each choice so that we can eliminate it. 

Let's look at the choice x+y > 0. We could let -0.75, and we could let y = 0.5, and this would satisfy the conditions for x and y. If x is -0.75 and y is 0.5, then x + y = -0.25 < 0, so this choice doesn't have to be true.

Let's then look at the choice -1 < xy < 1. If we let y be 10 and x be -0.5 then xy would be -5. This means that this choice isn't neccessarily true. 

Let's look at the choice (xy)2 > 1. We could let x be -0.5 and we could let y be 1. Then xy would be -0.5, and (xy)2 would be 0.25 < 1. This means that this choice isn't always true.

Let's look at the choice (x + y)2 > 1. We could let y = 0.5 and x = -0.5. Then x + y would equal 0, and (x+y)2 = 0 < 1. We can eliminate this choice as well.

We suspect that (x/y) < 1 might be the answer, because we can contradict every other statement. However, let's see if we can prove that (x/y) has to be less than 1. 

(x/y) < 1

We can multiply both sides by y, because y is positive, and this won't change the inequality sign. After multiplying both sides by y, we would have

x < y

Since y is always a positive number, and since x is always a negative number, this means that y will always be greater than x, so x < y must always be true. 

Example Question #1 : How To Find The Solution To An Inequality With Multiplication

If the inequality  is true, then which of the following must be true?

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

If we don't know anything for sure. 

could be less than , such as and .

could also be greater than , such as and .

In both of these cases, .

cannot be true.

The other three choices COULD be true, but do not HAVE to be true.

Example Question #2 : How To Find The Solution To An Inequality With Multiplication

Which of the following is the solution set of ?

Possible Answers:

Correct answer:

Explanation:

 

Remember that when you multiply both sides of the inequality by a negative number you must switch the inequality sign around (less than becomes greater than; greater than becomes less than)

Example Question #442 : Algebra

If –1 < n < 1, all of the following could be true EXCEPT:

Possible Answers:

(n-1)2 > n

|n2 - 1| > 1

n2 < n

16n2 - 1 = 0

n2 < 2n

Correct answer:

|n2 - 1| > 1

Explanation:

N_part_1

N_part_2

N_part_3

N_part_4

N_part_5

Example Question #222 : Algebra

(√(8) / -x ) <  2. Which of the following values could be x?

Possible Answers:

All of the answers choices are valid.

-1

-3

-4

-2

Correct answer:

-1

Explanation:

The equation simplifies to x > -1.41. -1 is the answer.

Example Question #443 : Algebra

Solve for x

\small 3x+7 \geq -2x+4

 

Possible Answers:

\small x \leq -\frac{3}{5}

\small x \geq \frac{3}{5}

\small x \leq \frac{3}{5}

\small x \geq -\frac{3}{5}

Correct answer:

\small x \geq -\frac{3}{5}

Explanation:

\small 3x+7 \geq -2x+4

\small 3x \geq -2x-3

\small 5x \geq -3

\small x\geq -\frac{3}{5}

Example Question #224 : Algebra

We have , find the solution set for this inequality. 

Possible Answers:

Correct answer:

Explanation:

Learning Tools by Varsity Tutors