[PSAT] 2-MONTH FINAL PREP

2-Month Final Prep

SATashkent

Expressions

Real-life application of expressions

Expressions are mathematical phrases that combine numbers, variables, and operations to model real-world scenarios. They serve as templates to calculate things like costs, distances, or quantities.


For instance, a total cost might include a fixed fee plus a variable rate.

Multiple Choice

A train travels 60 miles per hour. What is the distance it travels in 3 hours?

• 200 miles
• 120 miles
• 150 miles
• 180 miles

Multiple Choice

A store charges a $5 delivery fee plus $2 per item. What is the total cost for 8 items?

• 15
• 10
• 30
• 21

Simplify expressions

​Simplifying expressions means combining like terms
(same variable and exponent) and applying operations like distribution.
This reduces complexity while maintaining the expression’s value.


For example, 2x + 3x becomes 5x , and 4(x - 1) simplifies to 4x - 4.

Multiple Choice

Simplify: 6x - 2(x + 3) + x². What is the constant term?

• -3
• -6
• 0
• 6

Multiple Choice

Simplify: 3m + 4(m - 1) - 2m². What is the coefficient of m ?

• -2
• 7
• 5
• 4m

“Qisqa ko’paytirish formulalari”

Multiple Choice

If 49k² - 81 = (Ak - B)(Ak + B), find the value of A + B.

• -2

• 16

• 81

• 9

Multiple Choice

When the expression (5c + 4d)(25c² - 20cd + 16d²) is fully expanded, it takes the form Pc³ + Qd³. What is the value of P + Q?

• 189
• 150
• 175
• 200

Multiple Choice

The expression 16x² - 56xy + 49y² is the result of squaring a binomial. If this expression is equivalent to (Mx - Ny)², determine the value of M + N.

• 14
• 7
• 9
• 11

Equivalent expressions

Equivalent expressions produce the same result for any input, despite appearing different.

They can be confirmed by simplifying, expanding, or factoring.

For example, 3(x + 1) and 3x + 3 are equivalent.

Multiple Choice

If 4x + 12 = 4(x + k) , what is k?

• 3
• -3
• 5
• 1

Multiple Choice

Expand (x - 3)(x + 1) . What is the constant term?

• 1
• -4
• -3
• 3

Basic math terminologies(more, greater, less)

Words like “more” mean addition (e.g., “7 more than x ” is x + 7 ),
“greater” indicates comparison (e.g., x > 5 ), and
“less” means subtraction (e.g., “2 less than y ” is y - 2 ).


These terms help build expressions from descriptions.

Multiple Choice

What is the value of an expression that is 5 more than twice x when x = 3 ?

• 11
• 10
• 8
• 6

Multiple Choice

A quantity is 8 less than four times z . What is the value when z = 2?

• 0
• -2
• 4
• 6

Converting word problem into expression

To convert word problems into expressions, identify key quantities, relationships, and operations in the text.

This might involve combining fixed and variable terms or using geometric formulas. The result represents the situation mathematically

Multiple Choice

A club has a $30 signup fee plus $12 per event. What is the total cost after 5 events?

• 100
• 75
• 90
• 60

Multiple Choice

A baker is making cookies. He started with 'c' chocolate chips. He used 5 fewer chocolate chips than he started with for the first batch of cookies. For the second batch, he added 10 more chocolate chips than he used for the first batch.

Write an expression that represents the total number of chocolate chips the baker used for both batches of cookies.

• c + 5
• 2c
• c + 10
• c - 5

Linear Equations

Real-life application of linear equations

Linear equations model everyday situations with a fixed relationship between variables. They help calculate unknowns like costs or distances based on a constant rate.

For example, total expenses can be found when some costs are known and others vary.

Multiple Choice

A store charges 5 dollars for a bag plus 2 dollars per item. Write an equation for the total cost C for n items, and find the cost for 3 items.

• 5+2x;
  11

• 2x+5;

  15

• 5+2x;
  10

• 2x+5;

  8

Multiple Choice

You earn 10 dollars per hour. Write an equation for your earnings E after h hours, and calculate your earnings after 4 hours.

• E(h)=10h;

  60 dollars

• E(h)=10+10h;

  20 dollars

• E(h)=10+10h;

  80 dollars

• E(h)=10h;

  40 dollars

Solving linear equations

Solving linear equations means finding the variable’s value that makes the equation true. Use steps like isolating the variable, combining like terms, and applying inverse operations.


For example, in 2x - 4 = 6, add 4 and divide by 2 to get x = 5.

Multiple Choice

Solve for x: 5x + 2 = 3x - 8

• 0
• 4
• -2
• -5

Multiple Choice

Solve for y: 4(y - 1) = 2y + 6

• 7
• 3
• -1
• 5

Make subject questions

Making a variable the subject means rearranging an equation to express it in terms of other variables.
This isolates the desired variable using basic operations.


For example, in 2a = b - 3, divide by 2 to get a = (b - 3)/2.

Multiple Choice

Given 5a - b = 2, express a in terms of b.

• (b + 2) / 5
• (2 - b) / 5
• 5b - 2
• (b - 2) / 5

Multiple Choice

If 2x + 3y = 9, express y in terms of x.

• y = 9 - 2x
• y = 3x + 9
• y = 2x / 3
• y = (9 - 2x) / 3

Cross-multiplication

Multiple Choice

Solve for x: $$\frac{\left(x-1\right)}{3}=\frac{2}{5}$$

• 11/5
• 2/5
• 5/3
• 3/5

Multiple Choice

Solve for z: $$\frac{\left(3z+2\right)}{4}=\frac{1}{2}$$

• 2
• -1
• 0
• 1

Converting word problem in to linear equations

This means turning verbal descriptions into equations by assigning variables to unknowns and using given relationships.


For example, “a number plus twice another is 10” becomes x + 2y = 10. Solve the equation to find the values.

Multiple Choice

A number is added to three times another number. The result is 14.

If the number added is 2 more than the other number, what is the value of the number?

• 3
• 5
• 7
• 10

Multiple Choice

A trip costs 30 dollars for a ticket plus 5 dollars per hour. If the total cost is 45 dollars, write an equation to find the hours h and solve it.

• 2
• 4
• 3
• 5

Linear Inequalities

Real-life application of linear inequalities

Linear inequalities model situations with constraints, like budgeting or resource limits.

For example, they help determine how many items you can buy within a budget.

Multiple Choice

You have 50 dollars to spend on books that cost 12 dollars each. Write an inequality for the number of books you can buy and find the maximum number.

• 12x ≤ 50;

  maximum 3 books

• 12x ≤ 50;

  maximum 4 books

• 12x ≥ 50;

  maximum 5 books

• 50x ≤ 12;

  maximum 4 books

Multiple Choice

A car rental charges 30 dollars per day plus 0.20 dollars per mile. With a 100 dollar budget for one day, write an inequality for the miles you can drive and find the maximum miles.

• 30 + 0.20m ≤ 100;

  maximum 300 miles

• 30 + 0.20m ≤ 100;

  maximum 350 miles

• 0.20m ≤ 100;

  maximum 500 miles

• 30 + 0.20m ≥ 100;

  maximum 350 miles

Solving linear inequalities

Isolate the variable using steps similar to equations, but reverse the inequality sign when multiplying or dividing by a negative number.


The solution is a range of values.

Multiple Choice

Solve: 2x - 3 > 5

• x > 4
• x < 4
• x = 4
• x > 2

Multiple Choice

Solve: -3x + 4 ≤ 10

• x ≤ -3
• x ≥ -2
• x < -2
• x = -2

Knowing the meaning of inequality signs and their usage

Inequality signs (<, >, ≤, ≥) show relationships: less than, greater than, less than or equal to, greater than or equal to.


Inequality Signs and Their Meanings

< (Less Than)

Shows that one value is smaller than another

The variable can be any number below the given value

> (Greater Than)

Shows that one value is larger than another

The variable can be any number above the given value

≤ (Less Than or Equal To)

Shows that one value is smaller than or exactly equal to another

The variable can be any number below or equal to the given value

≥ (Greater Than or Equal To)

Shows that one value is larger than or exactly equal to another

The variable can be any number above or equal to the given value

Key Point:

- Symbols with lines underneath (≤, ≥) include the boundary number

- Symbols without lines (<, >) exclude the boundary number

Multiple Choice

What does x > 3 mean? Give an example.

• Closed circle at 3, shading to the left.

• Closed circle at 3, shading to the right.

• Open circle at 3, shading to the left.

• Open circle at 3, shading to the right.

Multiple Choice

Explain the difference between x ≥ 5 and x > 5 with examples.

• x ≥ 5 is always greater than x > 5.
• x ≥ 5 includes 5, while x > 5 does not.
• x ≥ 5 excludes 5, while x > 5 includes 5.
• x ≥ 5 and x > 5 are equivalent expressions.

Grid-in the solution in number-line

Represent the solution by shading the appropriate region on a number line, using open circles for < or > and closed circles for ≤ or ≥.

Multiple Choice

For the inequality x < 2, should the circle at 2 be open or closed, and in which direction should the shading go?

• Open circle at 2, shading to the left.
• Closed circle at 2, shading to the right.
• Open circle at 2, shading to the right.
• Closed circle at 2, shading to the left.

Multiple Choice

For the inequality x ≥ -1, should the circle at -1 be open or closed, and in which direction should the shading go?

• Open circle at -1, shading to the left.
• Closed circle at -1, shading to the left.
• Open circle at -1, shading to the right.
• Closed circle at -1, shading to the right.

Memorizing all the inequality-related vocabulary and their usage

Learn terms like

"at least" (≥), "at most" (≤), "more than" (>), "less than" (<),

to translate words into inequalities.

Multiple Choice

Translate "x is at least 5" into an inequality.

• x >= 5
• x < 5
• x = 5
• x > 5

Multiple Choice

Translate "y is no more than 10" into an inequality.

• y = 10
• y < 10
• y >= 10
• y <= 10

Converting word problem into linear inequality

Identify unknowns and constraints from the problem, then write an inequality using variables and appropriate signs.

Multiple Choice

A company has 500 dollars for supplies, each costing 25 dollars. Write an inequality for the number of items they can buy.

• 500 <= 25x
• 25x < 500
• x <= 20
• 25x <= 500

Multiple Choice

A tank holds up to 100 gallons. It has 30 gallons, and water is added at 5 gallons per minute. Write an inequality for the minutes they can add water without overflowing.

• m ≤ 10
• m ≤ 5
• m ≤ 14
• m ≤ 20