Equations & Inequalities

Question 1

What are the steps to solve problems involving unknown quantities?

Answer 1

1. Identify the unknown
2. Form an equation
3. Solve the equation
4. Interpret the solution and answer the question.

Question 2

What method is used to solve linear equations?

Answer 2

Balance method.

Question 3

What is the first step in solving the equation 5(x - 2) = 3x - 7?

Answer 3

Expand the equation to 5x - 10 = 3x - 7.

Question 4

What is the next step after 5x - 10 = 3x - 7?

Answer 4

Add 10 to both sides to get 5x = 3x + 3.

Question 5

In the equation 5x = 3x + 3, what do you do next to isolate x?

Answer 5

Subtract 3x from both sides to get 2x = 3.

Question 6

What is the solution for x in the equation 2x = 3?

Answer 6

x = 1.5.

Question 7

How do you solve linear inequalities?

Answer 7

In a similar way to linear equations.

Question 8

What does the inequality 2 - 2x < 6(x + 3) simplify to?

Answer 8

-16 < 8x.

Question 9

What is the solution for x in the inequality 8x > 16?

Answer 9

x > 2.

Question 10

What is the perimeter of the square if it is given as 28 cm?

Answer 10

The area is 196 cm².

Question 11

If the length of a rectangular room is 2m longer than its width and the perimeter is 16m, what are the dimensions?

Answer 11

Width = 4m, Length = 6m.

Question 12

Solve the equation 3x - 1 = x + 5.

Answer 12

x = 3.

Question 13

Solve the equation 6(x + 2) = 5(x + 4).

Answer 13

x = 2.

Question 14

Solve the equation 7 - 2x = 4x - 5.

Answer 14

x = 2.

Question 15

Find the solution to the inequality 4 + 3x < 13.

Answer 15

x < 3.

Question 16

Find the solution to the inequality 4x - 3 > 13.

Answer 16

x > 4.

Question 17

Find the solution to the inequality 3 - 2x ≤ 5 - 6x.

Answer 17

x ≤ 1.

Question 18

Find the solution to the inequality 5 + 7x > 10 + x.

Answer 18

x > 0.833.

Question 19

What is the elimination method used for?

Answer 19

Solving simultaneous linear equations

It involves manipulating equations to eliminate one variable, making it easier to solve for the other.

Question 20

What is the first step in solving the equations 3x - y = 1 and 2x + 3y = 8 using the elimination method?

Answer 20

Multiply the first equation by 3

This results in the equation 9x - 3y = 3.

Question 21

After multiplying the first equation by 3, what do you do next?

Answer 21

Add the new equation to the second equation

This combines 9x - 3y = 3 with 2x + 3y = 8.

Question 22

What is the resulting equation after adding 9x - 3y = 3 and 2x + 3y = 8?

Answer 22

11x = 11

Solving for x gives x = 1.

Question 23

What value of y is found by substituting x = 1 back into the equation 3x - y = 1?

Answer 23

y = 2

This completes the solution for the simultaneous equations.

Question 24

What method is used to find approximate solutions to equations when exact solutions are difficult?

Answer 24

Trial and improvement

This method involves testing various values to find a solution within a specified range.

Question 25

Using trial and improvement, what is the approximate solution to the equation x² - 10x = 18 between x = 0 and x = 12?

Answer 25

x = 11.6 to 1 decimal place ## Footnote This was determined by evaluating several values and finding one that gives a result closest to 18.

Question 26

What is the first equation that needs to be solved using the elimination method in the example provided?

Answer 26

2x + 3y = 7 ## Footnote This equation pairs with x - y = 1 for elimination.

Question 27

What is the second equation that needs to be solved using the elimination method in the example provided?

Answer 27

x - y = 1 ## Footnote This equation pairs with 2x + 3y = 7.

Question 28

What is the equation to be solved using trial and improvement for the solution between x = 0 and x = 4?

Answer 28

x + 9x = 32 ## Footnote This equation is part of the trial and improvement examples.

Question 29

What is the equation to be solved using trial and improvement for the solution between y = 2 and y = 6?

Answer 29

2y² + 3y = 50 ## Footnote This is another trial and improvement example.

Question 30

For the equation 3x² - 5x - 15 = 0, what is the range for the solution?

Answer 30

Between x = 3 and x = 1 ## Footnote This indicates where the solution can be found using trial and improvement.

Question 31

What is the equation to be solved using trial and improvement for the solution between x = 0 and x = 5?

Answer 31

2x + 3x² - 10x = 15 ## Footnote This equation is also part of the trial and improvement examples.