How to find the solution to an inequality with division

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Algebra › How to find the solution to an inequality with division

Questions 1 - 10

Solve for :

None of the other answers

Explanation

To solve for , separate the integers and 's by adding 1 and subtracting from both sides to get . Then, divide both sides by 2 to get . Since you didn't divide by a negative number, the sign does not need to be reversed.

Solve:

$x\geq-5$

Explanation

We will need to isolate the term with the x variable.

First subtract three from both sides.

Simplify both sides of the inequality.

Divide by a negative three on both sides. This will change switch the sign.

$\frac{-3x}{-3}>\frac{-15}{-3}$

Simplify both sides.

The answer is:

Find the solution set of the compound inequality:

$0.3x + 5 > 17 \textrm{ or } 0.7x - 3.1 < 1.6$

$(-\infty ,7) \cup (40,\infty )$

$(7,\infty )$

$(40,\infty )$

$(-\infty ,40 )$

Explanation

Solve each inequality separately:

$0.3x \div 0.3 > 12 \div 0.3$

or, in interval form, $(40,\infty )$

$0.7x \div 0.7 < 4.9 \div 0.7$

or, in interval form, $(-\infty ,7)$

Since these statements are connected by an "or", we are looking for the union of the intervals. Since the intervals are disjoint, we can simply write this as $(-\infty ,7) \cup (40,\infty )$

Solve the following inequality for :

Explanation

First, using the additive inverses, isolate the variables and constants on either side of the equation. In other words, subtract from both sides, and subtract from both sides. This leaves you with:

Now use the multiplicative inverse (divide by ) to solve the inequality.

Keep in mind that we do not flip the inequality symbol because the number we were dividing by is positive.

Solve the inequality:

$x\leq20$

$x \geq 20$

Explanation

Divide by negative three on both sides.

$\frac{-3x}{-3}>\frac{-60}{-3}$

Remember that since we are dividing by a negative sign, we will need to switch the sign.

The answer is:

Find the solution of the inequality: $-4x<\frac{3}{4}$

$x>-\frac{3}{16}$

$x<-\frac{3}{16}$

$x>-\frac{16}{3}$

Explanation

To isolate the x-variable, we will need to divide by negative four on both sides. This is the same as multiplying negative one fourth on both sides.

Since we have a negative in front, we will also need to switch the sign.

$-4x\cdot -\frac{1}{4}<\frac{3}{4}\cdot -\frac{1}{4}$

Simplify both sides of the equation and switch the sign.

$x>-\frac{3}{16}$

The answer is: $x>-\frac{3}{16}$

Give the solution set of the inequality:

$(-\infty ,-16)$

$(-\infty ,16)$

$\left (-16, \infty \right )$

The set of all real numbers

Explanation

$-4x \div (-4)< 64 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

or, in interval form,

$(-\infty ,-16)$

Solve the inequality.

$4x+3\geq -4$

$x\geq -\frac{7}{4}$

$x\leq -\frac{7}{4}$

$x> \frac{7}{4}$

$x> -\frac{7}{4}$

$x> \frac{3}{4}$

Explanation

Use the properties of inequalities to balance the inequality and isolate .

$4x+3\geq -4$

First subtract three from both sides.

$4x\geq -7$

Next, divide by four.

$x\geq -\frac{7}{4}$

Since no division or multiplication of a negative number occurred, the inequality sign remains the same.

Solve the inequality: $3x>\frac{1}{4}$

$x>\frac{1}{12}$

$x\leq \frac{1}{12}$

$x>\frac{3}{4}$

$x>\frac{4}{3}$

Explanation

Divide by three on both sides.

$\frac{3x}{3}>\frac{\frac{1}{4}}{3}$

Reduce both sides. For the right side, dividing by three is similar to multiplying by one-third.

$x>\frac{1}{4} \times \frac{1}{3}$

Simplify.

The answer is: $x>\frac{1}{12}$

Find the solution set to the following compound inequality statement:

$4y + 15 < 91 \textrm{ and } 7y -12 > 86$

$(14, \infty )$

$(-\infty , 19)$

$(-\infty , 14) \cup (19, \infty )$

$(-\infty , \infty )$

Explanation

Solve each of these two inequalities separately:

$4y \div 4 < 76 \div 4$

, or, in interval form, $(-\infty , 19)$

$7y \div 7 > 98 \div 7$

, or, in interval form, $(14, \infty )$

The two inequalities are connected with an "and", so we take the intersection of the two intervals.

$(14, \infty ) \cap (-\infty , 19) = (14,19)$

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