1.1E: Exercises - Solving Linear Equations and Inequalities

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Exercise $\PageIndex{1}$

Determine whether or not the given value is a solution.

$−5x + 4 = −1 ; x = −1$
$4x − 3 = −7 ; x = −1$
$3y − 4 = 5; y = \frac{9}{3}$
$−2y + 7 = 12 ; y = −\frac{5}{2}$
$3a − 6 = 18 − a; a = −3$
$5 (2t − 1) = 2 − t; t = 2$
$ax − b = 0; x = \frac{b}{a}$
$ax + b = 2b; x = \frac{b}{a}$

Answer

1. No

3. Yes

5. No

7. Yes

Exercise $\PageIndex{2}$

Solve.

$5x − 3 = 27$
$6x − 7 = 47$
$4x + 13 = 35$
$6x − 9 = 18$
$9a + 10 = 10$
$5 − 3a = 5$
$−8t + 5 = 15$
$−9t + 12 = 33$
$\frac{2}{3} x + \frac{1}{2} = 1$
$\frac{3}{8} x + \frac{5}{4} = \frac{3}{2}$
$\frac{1 − 3y}{5} = 2$
$\frac{2 − 5y}{6} = −8$
$7 − y = 22$
$6 − y = 12$
Solve for $x: ax − b = c$
Solve for $x: ax + b = 0$

Answer

1. $6$

3. $\frac{11}{2}$

5. $0$

7. $−\frac{5}{4}$

9. $\frac{3}{4}$

11. $−3$

13. $−15$

15. $x = \frac{b+c}{a}$

Exercise $\PageIndex{3}$

Solve.

$6x − 5 + 2x = 19$
$7 − 2x + 9 = 24$
$12x − 2 − 9x = 5x + 8$
$16 − 3x − 22 = 8 − 4x$
$5y − 6 − 9y = 3 − 2y + 8$
$7 − 9y + 12 = 3y + 11 − 11y$
$3 + 3a − 11 = 5a − 8 − 2a$
$2 − 3a = 5a + 7 − 8a$
$\frac{1}{3} x −\frac{3}{2} + \frac{5}{2} x = \frac{5}{6} x + \frac{1}{4}$
$\frac{5}{8} + \frac{1}{5} x −\frac{3}{4} = \frac{3}{10} x − \frac{1}{4}$
$1.2x − 0.5 − 2.6x = 2 − 2.4x$
$1.59 − 3.87x = 3.48 − 4.1x − 0.51$
$5 − 10x = 2x + 8 − 12x$
$8x − 3 − 3x = 5x − 3$
$5 (y + 2) = 3 (2y − 1) + 10$
$7 (y − 3) = 4 (2y + 1) − 21$
$7 − 5 (3t − 9) = 22$
$10 − 5 (3t + 7) = 20$
$5 − 2x = 4 − 2 (x − 4)$
$2 (4x − 5) + 7x = 5 (3x − 2)$
$4 (4a − 1) = 5 (a − 3) + 2 (a − 2)$
$6 (2b − 1) + 24b = 8 (3b − 1)$
$\frac{2}{3} (x + 18) + 2 = \frac{1}{3} x − 13$
$\frac{2}{5} x − \frac{1}{2} (6x − 3) = \frac{4}{3}$
$1.2 (2x + 1) + 0.6x = 4x$
$6 + 0.5 (7x − 5) = 2.5x + 0.3$
$5 (y + 3) = 15 (y + 1) − 10y$
$3 (4 − y) − 2 (y + 7) = −5y$
$\frac{1}{5} (2a + 3) −\frac{1}{2} = \frac{1}{3} a + \frac{1}{10}$
$\frac{3}{2} a = \frac{3}{4} (1 + 2a) −\frac{1}{5} (a + 5)$
$6 − 3 (7x + 1) = 7 (4 − 3x)$
$6 (x − 6) − 3 (2x − 9) = −9$
$\frac{3}{4} (y − 2) + \frac{2}{3} (2y + 3) = 3$
$\frac{5}{4} − \frac{1}{2} (4y − 3) = \frac{2}{5} (y − 1)$
$−2 (3x + 1) − (x − 3) = −7x + 1$
$6 (2x + 1) − (10x + 9) = 0$
Solve for $w: P = 2l + 2w$
Solve for $a: P = a + b + c$
Solve for $t: D = rt$
Solve for $w: V = lwh$
Solve for $b: A = \frac{1}{2} bh$
Solve for $a:s = \frac{1}{2}at^{2}$
Solve for $a: A = \frac{1}{2}h (a + b)$
Solve for $h: V = \frac{1}{3}πr^{2}h$
Solve for $F: C = \frac{5}{9} (F − 32)$
Solve for $x: ax + b = c$

Answer

1. $3$

3. $−5$

5. $−\frac{17}{2}$

7. $ℝ$

9. $\frac{7}{8}$

11. $2.5$

13. $Ø$

15. $3$

17. $2$

19. $Ø$

21. $−\frac{5}{3}$

23. $−81$

25. $1.2$

27. $ℝ$

29. $0$

31. $Ø$

33. $\frac{6}{5}$

35. $ℝ$

37. $w = \frac{P − 2l}{2}$

39. $t = \frac{D}{r}$

41. $b = \frac{2A}{h}$

43. $a = \frac{2A}{h} − b$

45. $F = \frac{9}{5} C + 32$

Exercise $\PageIndex{4}$

Set up an algebraic equation then solve.

Number Problems

When $3$ is subtracted from the sum of a number and $10$ the result is $2$. Find the number.
The sum of $3$ times a number and $12$ is equal to $3$. Find the number.
Three times the sum of a number and $6$ is equal to $5$ times the number. Find the number.
Twice the sum of a number and $4$ is equal to $3$ times the sum of the number and $1$. Find the number.
A larger integer is $1$ more than $3$ times another integer. If the sum of the integers is $57$, find the integers.
A larger integer is $5$ more than twice another integer. If the sum of the integers is $83$, find the integers.
One integer is $3$ less than twice another integer. Find the integers if their sum is $135$.
One integer is $10$ less than $4$ times another integer. Find the integers if their sum is $100$.
The sum of three consecutive integers is $339$. Find the integers.
The sum of four consecutive integers is $130$. Find the integers.
The sum of three consecutive even integers is $174$. Find the integers.
The sum of four consecutive even integers is $116$. Find the integers.
The sum of three consecutive odd integers is $81$. Find the integers.
The sum of four consecutive odd integers is $176$. Find the integers.

Answer

1. $−5$

3. $9$

5. $14, 43$

7. $46, 89$

9. $112, 113, 114$

11. $56, 58, 60$

13. $25, 27, 29$

Exercise $\PageIndex{5}$

Geometry Problems

The length of a rectangle is $5$ centimeters less than twice its width. If the perimeter is $134$ centimeters, find the length and width.
The length of a rectangle is $4$ centimeters more than $3$ times its width. If the perimeter is $64$ centimeters, find the length and width.
The width of a rectangle is one-half that of its length. If the perimeter measures $36$ inches, find the dimensions of the rectangle.
The width of a rectangle is $4$ inches less than its length. If the perimeter measures $72$ inches, find the dimensions of the rectangle.
The perimeter of a square is $48$ inches. Find the length of each side.
The perimeter of an equilateral triangle is $96$ inches. Find the length of each side.
The circumference of a circle measures $80π$ units. Find the radius.
The circumference of a circle measures $25$ centimeters. Find the radius rounded off to the nearest hundredth.

Answer

1. Width: $24$ centimeters; length: $43$ centimeters

3. Width: $6$ inches; length: $12$ inches

5. $12$ inches

7. $40$ units

Exercise $\PageIndex{6}$

Simple Interest Problems

For how many years must $$1,000$ be invested at $5\frac{1}{2}$ % to earn $$165$ in simple interest?
For how many years must $$20,000$ be invested at $6\frac{1}{4}$ % to earn $$3,125$ in simple interest?
At what annual interest rate must $$6500$ be invested for $2$ years to yield $$1,040$ in simple interest?
At what annual interest rate must $$5,750$ be invested for $1$ year to yield $$333.50$ in simple interest?
If the simple interest earned for $5$ years was $$1,860$ and the annual interest rate was $6$%, what was the principal?
If the simple interest earned for $2$ years was $$543.75$ and the annual interest rate was $3\frac{3}{4}$ %, what was the principal?
How many years will it take $$600$ to double earning simple interest at a $5$% annual rate? (Hint: To double, the investment must earn $$600$ in simple interest.)
How many years will it take $$10,000$ to double earning simple interest at a $5$% annual rate? (Hint: To double, the investment must earn $$10,000$ in simple interest.)
Jim invested $$4,200$ in two accounts. One account earns $3$% simple interest and the other earns $6$%. If the interest after $1$ year was $$159$, how much did he invest in each account?
Jane has her $$6,500$ savings invested in two accounts. She has part of it in a CD at $5$% annual interest and the rest in a savings account that earns $4$% annual interest. If the simple interest earned from both accounts is $$303$ for the year, then how much does she have in each account?
Jose put last year’s bonus of $$8,400$ into two accounts. He invested part in a CD with $2.5$% annual interest and the rest in a money market fund with $1.5$% annual interest. His total interest for the year was $$198$. How much did he invest in each account?
Mary invested her total savings of $$3,300$ in two accounts. Her mutual fund account earned $6.2$% last year and her CD earned $2.4$%. If her total interest for the year was $$124.80$, how much was in each account?
Alice invests money into two accounts, one with $3$% annual interest and another with $5$% annual interest. She invests $3$ times as much in the higher yielding account as she does in the lower yielding account. If her total interest for the year is $$126$, how much did she invest in each account?
James invested an inheritance in two separate banks. One bank offered $5\frac{1}{2}$ % annual interest rate and the other $6\frac{1}{4}$%. He invested twice as much in the higher yielding bank account than he did in the other. If his total simple interest for $1$ year was $$5,760$, then what was the amount of his inheritance?

Answer

1. $3$ years

3. $8$%

5. $$6,200$

7. $20$ years

9. He invested $$3,100$ at $3$% and $$1,100$ at $6$%.

11. Jose invested $$7,200$ in the CD and $$1,200$ in the money market fund.

13. Alice invested $$700$ at $3$% and $$2,100$ at $5$%.

Exercise $\PageIndex{7}$

What is regarded as the main business of algebra? Explain.
What is the origin of the word algebra?
Create an identity or contradiction of your own and share it on the discussion board. Provide a solution and explain how you found it.
Post something you found particularly useful or interesting in this section. Explain why.
Conduct a web search for “solving linear equations.” Share a link to website or video tutorial that you think is helpful.

Answer

1. Answer may vary

3. Answer may vary

5. Answer may vary

Exercise $\PageIndex{8}$

Determine whether or not the given value is a solution.

$5 x - 1 < - 2 ; x = - 1$
$- 3 x + 1 > - 10 ; x = 1$
$2 x - 3 < - 5 ; x = 1$
$5 x - 7 < 0 ; x = 2$
$9 y - 4 \geq 5 ; y = 1$
$- 6 y + 1 \leq 3 ; y = - 1$
$12 a + 3 \leq - 2 ; a = - \frac { 1 } { 3 }$
$25 a - 2 \leq - 22 ; a = - \frac { 4 } { 5 }$
$- 10 < 2 x - 5 < - 5 ; x = - \frac { 1 } { 2 }$
$3 x + 8 < - 2 \text { or } 4 x - 2 > 5 ; x = 2$

Answer

1. Yes

3. No

5. Yes

7. No

9. Yes

Exercise $\PageIndex{9}$

Graph all solutions on a number line and provide the corresponding interval notation.

$3 x + 5 > - 4$
$2 x + 1 > - 1$
$5 - 6 y < - 1$
$7 - 9 y > 43$
$6 - a \leq 6$
$- 2 a + 5 > 5$
$\frac { 5 x + 6 } { 3 } \leq 7$
$\frac { 4 x + 11 } { 6 } \leq \frac { 1 } { 2 }$
$\frac { 1 } { 2 } y + \frac { 5 } { 4 } \geq \frac { 1 } { 4 }$
$\frac { 1 } { 12 } y + \frac { 2 } { 3 } \leq \frac { 5 } { 6 }$
$2 ( 3 x + 14 ) < - 2$
$5 ( 2 y + 9 ) > - 15$
$5 - 2 ( 4 + 3 y ) \leq 45$
$- 12 + 5 ( 5 - 2 x ) < 83$
$6 ( 7 - 2 a ) + 6 a \leq 12$
$2 a + 10 ( 4 - a ) \geq 8$
$9 ( 2 t - 3 ) - 3 ( 3 t + 2 ) < 30$
$- 3 ( t - 3 ) - ( 4 - t ) > 1$
$\frac { 1 } { 2 } ( 5 x + 4 ) + \frac { 5 } { 6 } x > - \frac { 4 } { 3 }$
$\frac { 2 } { 5 } + \frac { 1 } { 6 } ( 2 x - 3 ) \geq \frac { 1 } { 15 }$
$5 x - 2 ( x - 3 ) < 3 ( 2 x - 1 )$
$3 ( 2 x - 1 ) - 10 > 4 ( 3 x - 2 ) - 5 x$
$- 3 y \geq 3 ( y + 8 ) + 6 ( y - 1 )$
$12 \leq 4 ( y - 1 ) + 2 ( 2 y + 1 )$
$- 2 ( 5 t - 3 ) - 4 > 5 ( - 2 t + 3 )$
$- 7 ( 3 t - 4 ) > 2 ( 3 - 10 t ) - t$
$\frac { 1 } { 2 } ( x + 5 ) - \frac { 1 } { 3 } ( 2 x + 3 ) > \frac { 7 } { 6 } x + \frac { 3 } { 2 }$
$- \frac { 1 } { 3 } ( 2 x - 3 ) + \frac { 1 } { 4 } ( x - 6 ) \geq \frac { 1 } { 12 } x - \frac { 5 } { 4 }$
$4 ( 3 x + 4 ) \geq 3 ( 6 x + 5 ) - 6 x$
$1 - 4 ( 3 x + 7 ) < - 3 ( x + 9 ) - 9 x$
$6 - 3 ( 2 a - 1 ) \leq 4 ( 3 - a ) + 1$
$12 - 5 ( 2 a + 6 ) \geq 2 ( 5 - 4 a ) - a$

Answer

1. $( - 3 , \infty )$;

$a2d07ff66453c86a51eb9ee5ca94a731.png$

Figure 1.8.12

3. $( 1 , \infty )$;

5. $[ 0 , \infty )$;

7. $( - \infty , 3 ]$;

9. $[ - 2 , \infty )$;

11. $( - \infty , - 5 )$;

13. $[ - 8 , \infty )$;

15. $[ 5 , \infty )$;

17. $( - \infty , 7 )$;

19. $( - 1 , \infty )$;

21. $( 3 , \infty )$;

23. $\left( - \infty , - \frac { 3 } { 2 } \right]$;

25. $\emptyset$;

27. $( - \infty , 0 )$;

29. $\mathbb { R }$;

31. $[ - 2 , \infty )$;

Exercise $\PageIndex{10}$

Graph all solutions on a number line and provide the corresponding interval notation.

$- 1 < 2 x + 1 < 9$
$- 4 < 5 x + 11 < 16$
$- 7 \leq 6 y - 7 \leq 17$
$- 7 \leq 3 y + 5 \leq 2$
$- 7 < \frac { 3 x + 1 } { 2 } \leq 8$
$- 1 \leq \frac { 2 x + 7 } { 3 } < 1$
$- 4 \leq 11 - 5 t < 31$
$15 < 12 - t \leq 16$
$- \frac { 1 } { 3 } \leq \frac { 1 } { 6 } a + \frac { 1 } { 3 } \leq \frac { 1 } { 2 }$
$- \frac { 1 } { 6 } < \frac { 1 } { 3 } a + \frac { 5 } { 6 } < \frac { 3 } { 2 }$
$5 x + 2 < - 3 \text { or } 7 x - 6 > 15$
$4 x + 15 \leq - 1 \text { or } 3 x - 8 \geq - 11$
$8 x - 3 \leq 1 \text { or } 6 x - 7 \geq 8$
$6 x + 1 < - 3 \text { or } 9 x - 20 > - 5$
$8 x - 7 < 1 \text { or } 4 x + 11 > 3$
$10 x - 21 < 9 \text { or } 7 x + 9 \geq 30$
$7 + 2 y < 5 \text { or } 20 - 3 y > 5$
$5 - y < 5 \text { or } 7 - 8 y \leq 23$
$15 + 2 x < - 15 \text { or } 10 - 3 x > 40$
$10 - \frac { 1 } { 3 } x \leq 5 \text { or } 5 - \frac { 1 } { 2 } x \leq 15$
$9 - 2 x \leq 15 \text { and } 5 x - 3 \leq 7$
$5 - 4 x > 1 \text { and } 15 + 2 x \geq 5$
$7 y - 18 < 17 \text { and } 2 y - 15 < 25$
$13 y + 20 \geq 7 \text { and } 8 + 15 y > 8$
$5 - 4 x \leq 9 \text { and } 3 x + 13 \leq 1$
$17 - 5 x \geq 7 \text { and } 4 x - 7 > 1$
$9 y + 20 \leq 2 \text { and } 7 y + 15 \geq 1$
$21 - 6 y \leq 3 \text { and } - 7 + 2 y \leq - 1$
$- 21 < 6 ( x - 3 ) < - 9$
$0 \leq 2 ( 2 x + 5 ) < 8$
$- 15 \leq 5 + 4 ( 2 y - 3 ) < 17$
$5 < 8 - 3 ( 3 - 2 y ) \leq 29$
$5 < 5 - 3 ( 4 + t ) < 17$
$- 3 \leq 3 - 2 ( 5 + 2 t ) \leq 21$
$- 40 < 2 ( x + 5 ) - ( 5 - x ) \leq - 10$
$- 60 \leq 5 ( x - 4 ) - 2 ( x + 5 ) \leq 15$
$- \frac { 1 } { 2 } < \frac { 1 } { 30 } ( x - 10 ) < \frac { 1 } { 3 }$
$- \frac { 1 } { 5 } \leq \frac { 1 } { 15 } ( x - 7 ) \leq \frac { 1 } { 3 }$
$- 1 \leq \frac { a + 2 ( a - 2 ) } { 5 } \leq 0$
$0 < \frac { 5 + 2 ( a - 1 ) } { 6 } < 2$

Answer

1. $(- 1,4 )$;

3. $[0,4]$;

5. $(−5,5]$;

7. $(−4,3]$;

9. $[−4,1]$;

11. $(−∞,−1)∪(3,∞)$;

13. $(−∞,12]∪[52,∞)$;

15. $ℝ$;

17. $(−∞,5)$;

19. $(−∞,−10)$;

21. $[−3,2]$;

23. $(−∞,5)$;

25. $Ø$;

27. $−2$;

29. $(−12,32)$;

31. $[−1,3)$;

33. $(−8,−4)$;

35. $(−15,−5]$;

37. $(−5,20)$;

39. $[−13, 43]$;

Exercise $\PageIndex{11}$

Find all numbers that satisfy the given condition.

Three less than twice the sum of a number and $6$ is at most $13$.
Five less than $3$ times the sum of a number and $4$ is at most $10$.
Five times the sum of a number and $3$ is at least $5$.
Three times the difference between a number and $2$ is at least $12$.
The sum of $3$ times a number and $8$ is between $2$ and $20$.
Eight less than twice a number is between $−20$ and $−8$.
Four subtracted from three times some number is between $−4$ and $14$.
Nine subtracted from $5$ times some number is between $1$ and $11$.

Answer

1. $( - \infty , 2 ]$

3. $[ - 2 , \infty )$

5. $( - 2,4 )$

7. $( 0,6 )$

Exercise $\PageIndex{12}$

Set up an algebraic inequality and then solve.

With a golf club membership, costing $$120$ per month, each round of golf costs only $$35.00$. How many rounds of golf can a member play if he wishes to keep his costs $$270$ per month at most?
A rental truck costs $$95$ per day plus $$0.65$ per mile driven. How many miles can be driven on a one-day rental to keep the cost at most $$120$?
Mark earned $6, 7$, and $10$ points out of $10$ on the first three quizzes. What must he score on the fourth quiz to average at least $8$?
Joe earned scores of $78, 82, 88$ and $70$ on his first four algebra exams. What must he score on the fifth exam to average at least $80$?
A gymnast scored $13.2, 13.0, 14.3, 13.8$, and $14.6$ on the first five events. What must he score on the sixth event to average at least $14.0$?
A dancer scored $7.5$ and $8.2$ from the first two judges. What must her score from the third judge come in as if she is to average $8.4$ or higher?
If two times an angle is between $180$ degrees and $270$ degrees, then what are the bounds of the original angle?
The perimeter of a square must be between $120$ inches and $460$ inches. Find the length of all possible sides that satisfy this condition.
A computer is set to shut down if the temperature exceeds $45$°C. Give an equivalent statement using degrees Fahrenheit. Hint: $C = \frac{5}{9} (F − 32)$.
A certain antifreeze is effective for a temperature range of $−35$°C to $120$°C. Find the equivalent range in degrees Fahrenheit.

Answer

1. Members may play $4$ rounds or fewer.

3. Mark must earn at least $9$ points on the fourth quiz.

5. He must score a $15.1$ on the sixth event.

7. The angle is between $90$ degrees and $135$ degrees.

9. The computer will shut down when the temperature exceeds $113$°F.

Exercise $\PageIndex{13}$

Often students reverse the inequality when solving $5x + 2 < −18$? Why do you think this is a common error? Explain to a beginning algebra student why we do not.
Conduct a web search for “solving linear inequalities.” Share a link to website or video tutorial that you think is helpful.
Write your own $5$ key takeaways for this entire chapter. What did you find to be review and what did you find to be new? Share your thoughts on the discussion board.

Answer

1. Answer may vary

3. Answer may vary

Footnotes

¹³⁸Linear expressions related with the symbols $≤, <, ≥,$ and $>$ .

¹³⁹A real number that produces a true statement when its value is substituted for the variable.

¹⁴⁰Properties used to obtain equivalent inequalities and used as a means to solve them.

¹⁴¹Inequalities that share the same solution set.

¹⁴²Two or more inequalities in one statement joined by the word “and” or by the word “or.”