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Algebra 1 : Systems of Inequalities

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Example Questions

Example Question #21 : How To Find The Solution To An Inequality With Multiplication

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

Possible Answers:

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

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

x>4

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

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

Correct answer:

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

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

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Example Question #22 : Equations / Inequalities

Solve the following inequality: $\frac{1}{3}x + \frac{2}{3}>3$

Possible Answers:

x>1

x<7

x>7

$x>-\frac{7}{3}$

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

Correct answer:

x>7

Explanation:

In order to eliminate the fractions, multiply the inequality by three on both sides.

This will make the inequality easier to solve.

$3(\frac{1}{3}x + \frac{2}{3})>3(3)$

Distribute and simplify both sides.

x+2>9

Subtract two from both sides.

x+2-2>9-2

The answer is: x>7

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Example Question #23 : Equations / Inequalities

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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

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Example Question #22 : How To Find The Solution To An Inequality With Multiplication

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

Possible Answers:

$x\leq\frac{47}{7}$

$x\leq\frac{49}{2}$

$x<\frac{47}{7}$

$x<\frac{49}{2}$

$x>\frac{49}{2}$

Correct answer:

$x<\frac{49}{2}$

Explanation:

Solve this inequality by multiplying both sides by the reciprocal of the coefficient in front of the x-variable.

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

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

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

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Example Question #25 : Equations / Inequalities

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

Possible Answers:

x>10

x>100

x>16

x>4

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

Correct answer:

x>100

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: x>100

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Example Question #26 : Equations / Inequalities

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

Possible Answers:

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

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

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

x>-4

x>-2

Correct answer:

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

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

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Example Question #27 : Equations / Inequalities

Solve the inequality: $9x\leq -\frac{1}{3}$

Possible Answers:

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

$x< -\frac{1}{27}$

$x\leq -\frac{1}{27}$

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

$x\geq -\frac{1}{27}$

Correct answer:

$x\leq -\frac{1}{27}$

Explanation:

In order to isolate the x-variable, we will need to divide both sides by nine. To avoid writing a complex fraction on the right side, dividing by nine is similar to multiplying by one-ninth.

Multiply by one-ninth on both sides.

$9x \cdot \frac{1}{9}\leq -\frac{1}{3} \cdot \frac{1}{9}$

Simplify both sides. Multiply the denominator with denominator on the right side. Since we have divide by a positive number on both sides, there is no need to switch the sign.

The answer is: $x\leq -\frac{1}{27}$

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Example Question #21 : Systems Of Inequalities

Solve the inequality:

$\small x+12>4+3$

Possible Answers:

$\small x<-19$

$\small x>-5$

$\small x>19$

$\small x<5$

$\small x>11$

Correct answer:

$\small x>-5$

Explanation:

First combine like terms on the right side of the inequality to obtain $\small x+12>7$ . Next, try to isolate the variable: $\small x+12\mathbf{-12} >7\mathbf{-12}$ .

The answer is therefore $\small x>-5$ .

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Example Question #22 : Systems Of Inequalities

Solve the inequality:

$\small -(-x-4)\le6$

Possible Answers:

$\small x\le-2$

$\small x \ge2$

$\small x\le2$

$\small x\le-10$

$\small x\ge-2$

Correct answer:

$\small x\le2$

Explanation:

Distribute the negative sign first: $\small -(-x-4)\le6$ becomes $\small x+4\le6$ . Since there are no like-terms to combine on one side of the inequality sign, we will try to isolate the variable: $\small x+4\boldsymbol{-4} \le 6\boldsymbol{-4}$ . The answer is therefore $\small x\le2$ .

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Example Question #3 : How To Find The Solution To An Inequality With Subtraction

$\left | 4x+4 \right |< 20$

Which one of the following is is a valid value for ?

Possible Answers:

Correct answer:

Explanation:

Since the inequality includes absolute value, you have two possiblities to consider: when the outcome is positive and when it is negative. When you consider the negative outcome, you must flip the inequality sign to solve for :

|4x+4|<20

This means that 4x+4 is less than positive 20 AND greater than negative 20:

4x+4<20 AND 4x+4 > -20

For each case, you will first subtract 4 from the left to the right. Then, you will divide both sides by 4 to isolate :

4x<16 AND 4x>-24

x<4 AND x>-6

This gives you the interval for valid values of :

-6<x<4

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Algebra 1 : Systems of Inequalities

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #21 : How To Find The Solution To An Inequality With Multiplication

Example Question #22 : Equations / Inequalities

Example Question #23 : Equations / Inequalities

Example Question #22 : How To Find The Solution To An Inequality With Multiplication

Example Question #25 : Equations / Inequalities

Example Question #26 : Equations / Inequalities

Example Question #27 : Equations / Inequalities

Example Question #21 : Systems Of Inequalities

Example Question #22 : Systems Of Inequalities

Example Question #3 : How To Find The Solution To An Inequality With Subtraction

All Algebra 1 Resources

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