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

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!

Find the solution set for :

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

$\left [ - \frac{1}{4},\frac{7}{4} \right )$

$\left ( - \frac{1}{4},\frac{7}{4} \right ]$

$\left ( - \frac{7}{4},\frac{1}{4} \right ]$

$\left [ \frac{1}{4},\frac{7}{4} \right )$

$\left [ -\frac{7}{4},-\frac{1}{4} \right )$

Explanation

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

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

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, $\left [ - \frac{1}{4},\frac{7}{4} \right )$

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

Explanation

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:

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

$x\leq -36$

Explanation

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:

Solve the following inequality: $\frac{2}{5}x>40$

$x>\frac{35}{2}$

Explanation

Multiply the reciprocal of the fraction in front of the variable on both sides of the equation.

$\frac{2}{5}x \cdot \frac{5}{2}>40\cdot \frac{5}{2}$

Simplify both sides.

$x>\frac{200}{2}$

Reduce the fraction on the right side.

The answer is:

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

$x>\frac{49}{4}$

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

$x<\frac{49}{4}$

$x<\frac{45}{4}$

Explanation

To isolate the x variable, multiply the reciprocal of the fraction in front of the variable on both sides of the inequality.

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

Simplify both sides of the inequality.

The answer is: $x>\frac{49}{4}$

Solve for :

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

$\left ( \frac{15}{7},\infty \right )$

$\left ( \frac{13}{7},\infty \right )$

$\left ( \frac{11}{7},\infty \right )$

$\left ( \frac{5}{3},\infty \right )$

$\left ( \frac{7}{3},\infty \right )$

Explanation

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

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

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

Cross-cancel:

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

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

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

Solve the inequality: $\frac{2}{3}x >15$

$x>\frac{45}{2}$

$x<\frac{45}{2}$

$x<\frac{43}{3}$

$x>\frac{43}{3}$

$x>\frac{13}{3}$

Explanation

Multiply both sides by three halves.

$\frac{2}{3}x \cdot \frac{3}{2} >15 \cdot \frac{3}{2}$

Simplify both sides. Multiply the integer with the numerator on the right side of the inequality.

The answer is: $x>\frac{45}{2}$

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

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

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

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

Explanation

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

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

Simplify both sides. There is no need to switch the sign. Multiply the negative integer with the numerator on the right side of the inequality.

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

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

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

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

$x\leq2$

$x\geq2$

Explanation

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: $x<\frac{1}{2}$

How to find the solution to an inequality with multiplication

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