Algebra 1 : How to find the solution to an inequality with multiplication

Example Questions

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Example Question #1 : Equations / Inequalities

Solve the inequality:

Explanation:

In order to isolate the variable, we need to remove the coefficient. Since the operation between  and  is multiplication, we may want to divide both sides of the inequality by . Although this is a valid step, in order to simplify matters, instead of dividing by , we can multiplying by .

(Note: dividing by  is exactly the same as multplying by the reciprocal, .)

Thus,

and after multiplying and simplifying, we obtain .

Example Question #2 : Equations / Inequalities

Solve for :

Explanation:

Simplify  by combining like terms to get .

Then, add  and to both sides to separate the 's and intergers. This gives you .

Divide both sides to get . Since we didn't divide by a negative number, there is no need to reverse the sign.

Example Question #3 : Equations / Inequalities

Solve for :

Explanation:

Cross-cancel:

or, in interval form, .

Example Question #4 : Systems Of Inequalities

Solve for

The inequality has no solution.

Explanation:

Eliminate fractions by multiplying by the least common denominator - .

Cross-cancel:

Example Question #4 : Equations / Inequalities

Find the solution set for :

Explanation:

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

Cross-cancel:

or, in interval form,

Example Question #5 : Equations / Inequalities

Solve for :

None of the other answers are correct.

Explanation:

Subtract 4 from both sides.  Then subtract 9x:

Next divide both sides by -6.  Don't forget to switch the inequality because of the negative sign!

Example Question #6 : Equations / Inequalities

Solve for :

None of the other answers are correct.

Explanation:

To solve the inequality, subtract and add 12 to both sides to separate the from the integers:

Divide both sides by 2:

Note: The inequality sign is only flipped when dividing by negative numbers.

Example Question #7 : Equations / Inequalities

Solve for :

None of the other answers are correct.

Explanation:

First, combine the like terms on the righthand side of the inequality to get .

Then, subtract  and  from both sides to get .

Finally, divide both sides by :

Example Question #8 : Equations / Inequalities

Solve this inequality.

Explanation:

Isolate all the terms with  on one side and the other terms on the other side and solve for

First subtract the an x from both sides of the inequality. The subtract 2 from each side. This results in the following inequality.

Here, we need to divide both sides by . However, whenever we divide or multiply and inequality by a negative number, we have to also change the direction of the inequality.

.

Example Question #9 : Equations / Inequalities

Find all of the solutions to this inequality.

Explanation:

To solve an inequality, isolate the variable on one side with all other constants on the other side. To accomplish this, perform opposite operations to manipulate the inequality.

First, isolate the x by multiplying each side by two.

Whatever you do to one side you must also do to the other side.

This gives you: