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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #335 : Basic Single Variable Algebra

Solve the inequality: $-\frac{1}{3}x < 2x-1$

Possible Answers:

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

$x>-\frac{3}{5}$

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

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

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

Correct answer:

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

Explanation:

Add $\frac{1}{3}x$ on both sides of the inequality.

$-\frac{1}{3}x +\frac{1}{3}x< 2x-1+\frac{1}{3}x$

Simplify both sides.

$0<\frac{7}{3}x-1$

Add 1 on both sides.

$0+1<\frac{7}{3}x-1+1$

The inequality becomes:

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

Multiply the reciprocal of the coefficient in front of .

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

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

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

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Example Question #341 : Basic Single Variable Algebra

Solve the inequality: 8-3x>16+7x

Possible Answers:

$x<\frac{4}{5}$

x>2

$x<\frac{5}{4}$

$x>\frac{4}{5}$

$x<-\frac{4}{5}$

Correct answer:

$x<-\frac{4}{5}$

Explanation:

Add on both sides.

8-3x+3x>16+7x+3x

The inequality becomes:

8>16+10x

Subtract 16 on both sides.

8-16>16+10x-16

-8>10x

Divide by 10 on both sides.

$\frac{-8}{10}>\frac{10x}{10}$

Reduce both fractions.

$-\frac{4}{5}>x$

The answer is: $x<-\frac{4}{5}$

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Example Question #342 : Basic Single Variable Algebra

Solve: 2(8x+3)>-3-(x+2)

Possible Answers:

$x<-\frac{7}{17}$

$x>-\frac{5}{17}$

$x<-\frac{11}{17}$

$x>\frac{7}{17}$

$x>-\frac{11}{17}$

Correct answer:

$x>-\frac{11}{17}$

Explanation:

Distribute on both sides of the inequality in order to eliminate the parentheses.

2(8x)+2(3)>-3-1(x)-1(2)

The inequality becomes:

16x+6>- 3-x-2

16x+6>-5-x

Add on both sides.

16x+6+x> -5-x+x

17x+6>-5

Subtract 6 from both sides.

17x+6-6>-5-6

17x>-11

Divide by 17 on both sides.

$\frac{17x}{17}>\frac{-11}{17}$

The answer is: $x>-\frac{11}{17}$

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Example Question #343 : Basic Single Variable Algebra

Solve the inequality: $-3x-9\leq 6$

Possible Answers:

$x\leq 1$

$x\leq- 1$

$x\geq- 1$

$x\geq -5$

$x\leq -5$

Correct answer:

$x\geq -5$

Explanation:

Add 9 on both sides.

$-3x-9+9\leq 6+9$

$-3x\leq15$

Divide by negative three on both sides. Dividing by a negative number will switch the sign.

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

The answer is: $x\geq -5$

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Example Question #344 : Basic Single Variable Algebra

Solve the inequality: $\frac{2}{3}x+\frac{1}{4}\leq \frac{1}{6}$

Possible Answers:

$x\leq- 8$

$x\leq \frac{5}{8}$

$x\leq -\frac{1}{8}$

$x\leq \frac{8}{5}$

$x\leq \frac{1}{8}$

Correct answer:

$x\leq -\frac{1}{8}$

Explanation:

Multiply both sides by the least common denominator to eliminate the fractions. The LCD is 12.

$12(\frac{2}{3}x+\frac{1}{4})\leq \frac{1}{6} \cdot 12$

$8x+3\leq 2$

Subtract three on both sides.

$8x+3-3\leq 2-3$

$8x\leq -1$

Divide both sides by 8.

$\frac{8x}{8}\leq \frac{-1}{8}$

The answer is: $x\leq -\frac{1}{8}$

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Example Question #111 : Solving Inequalities

Solve: -8x-7>4(-3x+2)

Possible Answers:

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

$x<\frac{15}{4}$

$x>-\frac{15}{4}$

$x>\frac{15}{4}$

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

Correct answer:

$x>\frac{15}{4}$

Explanation:

Distribute the four through the binomial of the right side.

-8x-7>4(-3x)+4(2)

-8x-7>-12x+8

Add 12x and on both sides.

-8x-7+(12x)+[7]>-12x +(12x)+8+[7]

4x>15

Divide by four on both sides.

$\frac{4x}{4}>\frac{15}{4}$

The answer is: $x>\frac{15}{4}$

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Example Question #112 : Solving Inequalities

Solve.

-x+4>2x+2

Possible Answers:

$x<\frac{2}{3}$

$x<\frac{3}{2}$

No solution.

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

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

Correct answer:

$x<\frac{2}{3}$

Explanation:

Solve.

-x+4>2x+2

Step 1: Subtract from both sides of the inequality.

Step 2: Subtract from both sides of the inequality to isolate the term with the variable.

Step 3: Multiply both sides of the inequality by -1 and reverse the inequality sign.

This is to make the inequality have only positive numbers, and this will help solve the inequality. Because we are multiplying by a negative number, we must reverse the inequality sign. The only times we reverse the inequality sign are when we are multiplying or dividing by a negative number. In other instances, we would leave the sign the same.

3x<2

Step 4: Divide both sides of the inequality by .

$x<\frac{2}{3}$

Solution: $x<\frac{2}{3}$

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Example Question #113 : Solving Inequalities

Solve the inequality for

$-2\leq \frac{1-3x}{4}\leq 4$

Possible Answers:

$5\leq x \leq -7$

$3\leq x\leq -5$

$-5\leq x\leq3$

$-5\leq x \leq 7$

$3\leq x\leq 7$

Correct answer:

$-5\leq x\leq3$

Explanation:

$-2\leq \frac{1-3x}{4}\leq 4$

Inequalities can be algebraically rearranged using operations that are mostly identical to algebraic equations, although one notable exception is multiplication or division by -1. This reverses the inequality signs.

Multiply out by

$-8\leq \1-3x\leq 16$

Subtract from all sides,

$-9\leq -3x\leq 15$

Divide throughout by and remember to reverse the inequality signs.

$3\geq x\geq -5$

It feels more natural to write the final result as:

$-5\leq x\leq3$

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Example Question #111 : Solving Inequalities

Solve for m.

17-8(5-2m)>4m+7+6(3m-3)

Possible Answers:

$m<\frac{16}{27}$

m<2

m>-2

$m>-{\frac{16}{27}}$

m<-2

Correct answer:

m<-2

Explanation:

Remember: Use inverse operations to undo the operations in the inequality (for example use a subtraction to undo an addition) until you are left with the variable. Make sure to do the same operations to both sides of the inequality.

Important Note: When multiplying or dividing by a negative number, always flip the sign of an inequality.

Solution:

$17-8\times(5-2m)>4m+7+6\times(3m-3)$

Expand all factors

17-40+16m>4m+7+18m-18

Simplify

-23+16m>22m-11

Add 23

Subtract 22m

Divide by -6 (We flip the sign of the inequality)

$m<\frac{12}{-6}$

Simplify

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Example Question #2181 : Algebra Ii

Solve the double inequality and give the solution in interval notation.

$5\leq4x+1\leq13$

Possible Answers:

$\left[\frac{3}{2},\frac{7}{2}\right]$

[4,12]

[1,3]

(1,3)

$\left(\frac{3}{2},\frac{7}{2}\right)$

Correct answer:

[1,3]

Explanation:

Start by subtracting 1 and divinding by 4 on both sides of the equality

$4\leq4x\leq12$

$1\leq x\leq3$

Written in interval notation:

$\left[ 1,3\right]$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #335 : Basic Single Variable Algebra

Example Question #341 : Basic Single Variable Algebra

Example Question #342 : Basic Single Variable Algebra

Example Question #343 : Basic Single Variable Algebra

Example Question #344 : Basic Single Variable Algebra

Example Question #111 : Solving Inequalities

Example Question #112 : Solving Inequalities

Example Question #113 : Solving Inequalities

Example Question #111 : Solving Inequalities

Example Question #2181 : Algebra Ii

All Algebra II Resources

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