How to find the solution to an inequality with division - ACT Math

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Question

Solve for :

Answer

Begin by adding to both sides, this will get the variable isolated:

Or...

Next, divide both sides by :

$x > -\frac{27}{4}$

Notice that when you divide by a negative number, you need to flip the inequality sign!

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Question

Solve the following inequality:

Answer

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,

$3x>27\Rightarrow \frac{3x}{3}>\frac{27}{3}\Rightarrow x>9$

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Question

Each of the following is equivalent to

xy/z * (5(x + y)) EXCEPT:

Answer

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1. We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z. xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression. 5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out. Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

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Question

Find is the solution set for x where:

$\left | 5x-3\right |>12$

Answer

We start by splitting this into two inequalities, and

We solve each one, giving us or $x<\frac{-9}{5}\right$ .

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Question

Which of the following inequalities defines the solution set for the inequality 14 – 3_x_ ≤ 5?

Answer

To solve this inequality, you should first subtract 14 from both sides.

This leaves you with –3_x_ ≤ –9.

In the next step, you divide both sides by –3, remembering to flip the inequality sign when you do this.

This leaves you with the solution x ≥ 3.

If you selected x ≤ 3, you probably forgot to flip the sign. If you selected one of the other solutions, you may have subtracted incorrectly.

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Question

Solve 3 < 5x + 7

Answer

Subtract seven from both sides, then divide both sides by 5, giving you –4/5 < x.

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Question

Solve 6_x_ – 13 > 41

Answer

Add 13 to both sides, giving you 6_x_ > 54, divide both sides by 6, leaving x > 9.

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Question

What is the solution set of the inequality $\dpi{100} \small 3x+8<35$ ?

Answer

We simplify this inequality similarly to how we would simplify an equation

$\dpi{100} \small 3x+8-8<35-8$

$\dpi{100} \small \frac{3x}{3}<\frac{27}{3}$

Thus $\dpi{100} \small x<9$

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Question

Solve for the -intercept:

$3y+11\geq 5y+6x-1$

Answer

Don't forget to switch the inequality direction if you multiply or divide by a negative.

$3y+11\geq 5y+6x-1$

$-2y+11\geq6x-1$

$-2y\geq6x-12$

$-\frac{1}{2}(-2y\geq 6x-12)$

$y\leq -3x+6$

Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.

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Question

Solve for .

$\small 14-2x\geq 22$

Answer

$\small 14-2x\geq22$

$\small -2x\geq8$

When dividing both sides of an inequality by a negative number, you must change the direction of the inequality sign.

$\small x\leq-4$

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Question

What is a solution set of the inequality ?

Answer

In order to find the solution set, we solve as we would an equation:

Therefore, the solution set is any value of .

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Question

Which of the following could be a value of , given the following inequality?

$8-7x\leq11x+4$

Answer

The inequality that is presented in the problem is:

$8-7x\leq11x+4$

Start by moving your variables to one side of the inequality and all other numbers to the other side:

$-7x-11x\leq4-8$

$-18x\leq-4$

Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!

$x\geq\frac{-4}{-18}$

Reduce:

$x\geq\frac{2}{9}$

The only answer choice with a value greater than $\frac{2}{9}$ is $\frac{4}{9}$ .

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Question

Simplify the following inequality

$4 - 3x \ge 22 + 2x$ .

Answer

For the most part, you can treat inequalities just like equations. (It is not exact, as you will see below.) Thus, start by isolating your variables. Subtract from both sides:

$4 - 5x \ge 22$

Next, subtract from both sides:

$- 5x \ge 18$

Finally—here you need to be careful—divide by . When you divide or multiply by a negative value in inequalities, you need to flip the inequality sign.

Thus, you get:

$x \le \frac{-18}{5}$

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Question

What is the solution to the given inequality:
$-3x\geq 12$

Answer

When solving an inequality in which you have to mulitiply or divide by a negative number, you must "flip" the direction of the inequality. Other than that, solve it nomrally.

Thus the first and only step we have is to divide by and since that number is negative, we "flip" the inequality.

Yielding:

$\frac{-3x}{-3}\geq \frac{12}{-3}$

$x\leq -4$

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Question

Solve the following inequality:

Answer

To solve, simply solve as though it is an equation.

The goal is to isolate the variable on one side with all other constants on the other side. Perform the opposite operation to manipulate the inequality.

Only when dividing or multiplying by a negative number, will you have to flip the inequality sign.

$x=\frac{2}{4}$

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

$x>\frac{1}{2}$

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