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

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Question

Solve the inequality:

$\small \frac{5}{7}x > 15$

Answer

In order to isolate the variable, we need to remove the coefficient. Since the operation between $\tiny \small \frac{5}{7}$ and $\small x$ is multiplication, we may want to divide both sides of the inequality by $\tiny \small \frac{5}{7}$ . Although this is a valid step, in order to simplify matters, instead of dividing by $\tiny \small \frac{5}{7}$ , we can multiplying by $\small \frac{7}{5}$ .

(Note: dividing by $\frac{5}{7}$ is exactly the same as multplying by the reciprocal, $\frac{7}{5}$ .)

Thus,

$\small x> \frac{15}{1} \cdot\frac{7}{5}$ and after multiplying and simplifying, we obtain $\small x>21$ .

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Question

Solve for :

Answer

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!

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Question

Solve for :

Answer

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.

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Question

Find the solution set for :

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

Answer

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$- \frac{5}{2} \cdot \left (-\frac{7}{10} \right )> - \frac{5}{2} \cdot \left ( -\frac{2}{5}x \right ) \geq - \frac{5}{2} \cdot \frac{1}{10}$

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

$\frac{5}{2} \cdot \frac{7}{10} > x \geq - \frac{5}{2} \cdot \frac{1}{10}$

Cross-cancel:

$\frac{1}{2} \cdot \frac{7}{2} > x \geq - \frac{1}{2} \cdot \frac{1}{2}$

$\frac{7}{4} > x \geq - \frac{1}{4}$

or, in interval form, $\left [ - \frac{1}{4},\frac{7}{4} \right )$

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Question

Solve for :

Answer

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.

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Question

Solve for :

$\frac{2}{9} x + \frac{1}{3} < \frac{5}{6}$

Answer

Eliminate fractions by multiplying by the least common denominator - .

$18 \cdot \left (\frac{2}{9} x + \frac{1}{3} \right ) <18 \cdot \frac{5}{6}$

$\frac{18}{1} \cdot \frac{2}{9} x + \frac{18}{1} \cdot \frac{1}{3} \right ) <\frac{18}{1} \cdot \frac{5}{6}$

Cross-cancel:

$\frac{2}{1} \cdot \frac{2}{1} x + \frac{6}{1} \cdot \frac{1}{1} \right ) <\frac{3}{1} \cdot \frac{5}{1}$

$4x \div 4 < 9 \div 4$

$x < \frac{9}{4}$

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Question

Solve for :

$\frac{7}{9} x > \frac{5}{3}$

Answer

$\frac{7}{9} x > \frac{5}{3}$

$\frac{7}{9} x \div \frac{7}{9}> \frac{5}{3} \div \frac{7}{9}$

$\frac{7}{9} x \cdot \frac{9}{7}> \frac{5}{3}\cdot \frac{9}{7}$

Cross-cancel:

$x> \frac{5}{1}\cdot \frac{3}{7}$

$x> \frac{15}{7}$

or, in interval form, $\left ( \frac{15}{7},\infty \right )$ .

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Question

Solve for :

Answer

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 :

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Question

Solve this inequality.

Answer

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.

The final answer becomes

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

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Question

$2x\leq6$

Find all of the solutions to this inequality.

Answer

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.

$\frac{2x}{{\color{Blue} 2}}\leq\frac{6}{{\color{Blue} 2}}$

This gives you:

$x\leq3$

The answer, therefore, is $(-\infty ,3]$ .

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Question

Solve the following inequality: $\frac{x}{4}\leq4$

Answer

In order to isolate variable, multiply by four on both sides of the equation.

$\frac{x}{4}\cdot 4\leq4 \cdot 4$

Simplify both sides of the inequality.

$x\leq 16$

The answer is: $x\leq 16$

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Question

Solve: $\frac{x}{7}>7$

Answer

Multiply by seven on both sides of the equation. There is no need to change the direction of the sign unless there is a negative sign.

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

Simplify both sides.

The answer is:

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Question

Solve: $\frac{9x}{7}<3$

Answer

Multiply both sides by the reciprocal of the fraction in front of the variable.

$\frac{9x}{7} \times \frac{7}{9}<3 \times \frac{7}{9}$

Simplify both sides. The nine on the right side of the inequality can be split into factors.

$x<\frac{3}{1} \times \frac{7}{3\times 3}$

The answer is: $x<\frac{7}{3}$

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Question

Solve the inequality: $\frac{6}{7}x >42$

Answer

In order to isolate the unknown variable, we will need to multiply the reciprocal of the coefficient on both sides of the inequality sign.

$\frac{6}{7}x \cdot \frac{7}{6}>42\cdot \frac{7}{6}$

Simplify both sides of the equation.

$x>42\cdot \frac{7}{6}$

The answer is:

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Question

Solve the inequality and rewrite in interval notation: $\frac{3}{4}x \geq \frac{1}{3}$

Answer

In order to isolate the variable, we can multiply both sides by the reciprocal of the coefficient in front of the x variable.

$\frac{3}{4}x \cdot \frac{4}{3} \geq \frac{1}{3} \cdot \frac{4}{3}$

Simplify both sides of the equation.

$x\geq \frac{4}{9}$

This indicates that the x-variable is four ninths or greater. Use a bracket sign to indicate that it includes the fraction. The infinity isn't finite, which means that a parenthesis should enclose this symbol instead of a bracket.

The answer is: $[\frac{4}{9},\infty)$

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Question

Solve the inequality: $-\frac{x}{6}>6$

Answer

In order to isolate the x variable, we will need to divide by negative one sixth on both sides. The result will switch the sign of the inequality.

This is also the same as multiplying by negative six on both sides.

$-\frac{x}{6} \cdot -6>6 \cdot -6$

Switch the sign.

Upon testing values that are less than negative 36, we will find that those values will satisfy the inequality instead of .

The answer is:

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Question

Solve the following inequality: $\frac{2x}{7}\geq7$

Answer

In order to isolate the x-variable, we will need to multiply both sides by the reciprocal of the coefficient in front of the x.

$\frac{2x}{7}\cdot \frac{7}{2}\geq7 \cdot \frac{7}{2}$

Simplify both sides. A whole number will multiply to the numerator of a fraction.

The answer is: $x\geq\frac{49}{2}$

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Question

Solve: $\frac{1}{8}x > -4$

Answer

To isolate the x-variable, we will need to multiply by eight on both sides.

$\frac{1}{8}x \cdot 8> -4\cdot 8$

Simplify both sides of the equation.

The answer is:

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Question

Find the solution to the inequality: $\frac{x}{9}<\frac{1}{18}$

Answer

In order to isolate the x-variable, we will need to multiply by nine on both sides of the inequality.

$\frac{x}{9} \cdot 9<\frac{1}{18}\cdot 9$

Simplify both sides of the equation.

$x<\frac{9}{18}$

Reduce the fraction on the right side.

The answer is: $x<\frac{1}{2}$

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Question

Solve the inequality: $-\frac{2}{3}x< 5$

Answer

In order to isolate the x-variable, we will need to multiply by the negative reciprocal of two-thirds on both sides of the equation.

$-\frac{2}{3}x \cdot -\frac{3}{2}< 5 \cdot-\frac{3}{2}$

Simplify both sides of the inequality. Since we have a negative coefficient, and multiplying by the reciprocal is similar to dividing, we will need to switch the sign.

$x>-\frac{15}{2}$

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