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Translate relationships, reverse inequalities correctly and preserve feasible solutions.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Solve an unfamiliar linear model or system and enforce its inequality, integer and contextual restrictions.
Adding the same expression to both sides or multiplying both sides by the same nonzero number preserves an equation's solution set. Distributing a factor across parentheses changes its representation without changing its value. Dividing by a variable expression requires checking that it is not zero; if it can be zero, handle that possibility before dividing. Distinguish simplifying an expression from solving an equation: an expression alone does not assert an equality that fixes the variable.
| Term | What it means |
|---|---|
| Solution set | Every value satisfying the original relations and restrictions. |
| System | Equations or inequalities that must hold at the same time. |
| Feasible | Allowed by all mathematical and contextual conditions. |
| Strict inequality | An inequality that excludes equality at its boundary. |
Algebra replaces an unknown quantity with a symbol while keeping the stated relationships intact. Solving means finding all values that satisfy those relationships and the original domain. A final number is not automatically a valid answer merely because the last line of an algebraic manipulation produced it. A count must be a permitted integer, a length must be positive, and a solution to a system must satisfy every equation simultaneously.
Before manipulating, define the variable in words and units. Translate one relationship at a time, then compare the model back to the sentence. Four less than three times a number means 3x - 4, not 3(x - 4). The difference comes from which operation the subtraction modifies. Good translation is often the hardest part of an otherwise short calculation; slow down there before trying to save time on arithmetic.
Another way: steps
Define variables; translate each relationship; solve with equivalent operations; intersect restrictions; substitute into the original situation.
To solve 5(x - 3) = 20, divide both sides by five, then add three to both sides. Alternatively distribute first and collect terms. Both routes preserve exactly the same solutions. A useful check is to substitute the proposed answer into the original equation, where a distribution error remains visible. Avoid checking only a later line that already contains the mistake. If all variable terms cancel, inspect what remains: a true statement such as 4 = 4 means every allowed input satisfies the equation, while 4 = 7 means none does.
Adding the same value preserves order. Multiplying or dividing both sides by a positive value also preserves order, while a negative value reverses it. For instance, -2x < 6 becomes x > -3. Multiplication by -1 reflects the number line, which explains the reversal rather than making it an arbitrary rule to memorize. Never divide an inequality by a variable of unknown sign without separating positive, negative and zero cases. For a constant coefficient, determine its sign before you write the next line.
The condition -1 <= 2x + 3 < 9 is a pair of simultaneous inequalities. Subtract three throughout and divide by positive two to get -2 <= x < 3. If x is an integer, the permitted values are -2, -1, 0, 1 and 2. The right endpoint is excluded even though it is close to the interval. A question asking how many integers satisfy the condition wants a count, not the interval endpoints or their difference without checking inclusion. For an inclusive interval of integers from a through b, the count is b - a + 1.
Substitution works well when one equation already isolates a variable. If y = 2x + 1, replace every y in the other equation with that expression. Elimination works well when coefficients can be made opposites by small multipliers. Adding the resulting equations cancels one variable. After finding it, recover the other and verify both original equations. A system can have one solution, no solutions or infinitely many. Parallel distinct line equations produce a contradiction when eliminated; equivalent equations produce an identity. For example, x + y = 4 and 2x + 2y = 8 describe the same line and leave infinitely many real pairs. Replacing eight by nine makes the pair inconsistent. Counting equations alone does not establish that a system has a unique solution.
The equation |x - 2| = 5 says that x lies five units from two, so x can be seven or negative three. Writing only x - 2 = 5 loses a valid solution. A negative right side, such as |x - 2| = -5, gives no real solution because distance cannot be negative. For an inequality with positive bound c, |x - a| < c describes the interval a - c < x < a + c. For |x - a| > c, values lie outside that interval, in two separate regions. Keep the equality boundary consistent.
A system may ask for x + y rather than for each variable separately. If adding the equations directly isolates that sum, solve the target instead of introducing extra arithmetic. Likewise, a cost model may ask for the greatest affordable whole number rather than the exact real-valued boundary. Calculate the bound, then choose the largest allowed integer below or at it, as appropriate. Rounding to the nearest integer is wrong when the result would exceed a budget or fail a minimum requirement.
Suppose a ticket problem gives a total count and a total price. Define adult tickets as a and child tickets as c. The count equation a + c = total and the price equation adult_price times a plus child_price times c = revenue have different units and roles. Substituting c = total - a reduces the system, but you must still check both ticket counts are nonnegative integers. A mathematically solved system with a negative ticket count indicates inconsistent data or a mistranslation, not a valid purchase.
A laboratory buys supplies only in whole boxes. If a box contains six tubes and the laboratory needs at least forty tubes, the real-valued quotient forty divided by six is a lower bound on the number of boxes. The smallest feasible integer is seven, because six boxes provide only thirty-six tubes. If each box costs nine dollars, the least feasible total is sixty-three dollars. The requested answer is a cost, not a box count or a tube count.
This example joins an inequality, an integer constraint and a unit conversion. Keep them in that order: first express the required amount, then choose a feasible count, then calculate the requested cost. Rounding forty divided by six to the nearest whole number happens to give seven, but the general rule is to round upward for a minimum purchase. A requirement of thirty-seven tubes also needs seven boxes, even though the quotient is closer to six.
In review, identify whether a mistake arose in translation, algebra or the final feasibility decision. These failures require different repairs, and a calculator alone addresses none of the interpretation errors.
Reverse order when dividing an inequality by a negative quantity. Keep strict endpoints excluded. Retain both branches of an absolute-value equation when appropriate. A maximum affordable integer must not exceed the bound; a minimum number of boxes must actually meet the demand. Check both equations in a system and every original domain restriction.
Read the equation exactly.
4(2x - 1) = 28.
The factor four multiplies the entire parenthesis.
Undo the outer multiplication.
2x - 1 = 7.
Divide both sides by nonzero four.
Isolate the variable value.
2x = 8, so x = 4.
Add one, then divide by two.
Check in the original equation.
4(8 - 1) = 28.
The proposed value satisfies the original relation.
Read the complete condition.
9 - 4n < 1; n is an integer.
Find the smallest permitted n.
Remove the constant term.
-4n < -8.
Subtract nine from both sides.
Divide and reverse direction.
n > 2.
The divisor negative four reverses the inequality.
Apply the integer domain.
The least permitted integer is 3.
The strict boundary excludes two.
Verify the boundary cases.
At three the left side is -3; at two it is 1.
Three satisfies the strict inequality while two does not.
Define both unknown counts.
a adult tickets at 11 dollars; c child tickets at 6 dollars.
There are thirteen tickets costing 108 dollars.
Translate total count and cost.
a + c = 13; 11a + 6c = 108.
The first equation counts tickets and the second counts dollars.
Express one unknown from the count.
c = 13 - a.
Each adult ticket leaves one fewer child ticket in the total.
Substitute into the cost equation.
11a + 6(13 - a) = 108.
Use the same a consistently in both relationships.
Solve and recover the other count.
5a = 30, so a = 6 and c = 7.
Both counts are nonnegative integers.
Verify both original totals.
6 + 7 = 13; 66 + 42 = 108.
The pair satisfies the count and cost conditions simultaneously.
Write both simultaneous equations.
2x + y = 14; 3x - y = 11.
The coefficients of y already cancel on addition.
Add corresponding sides.
5x = 25.
Solve for x, then recover y from either original equation.
Verify the ordered pair.
If 3(x - 2) = 21, what is x?
If x + y = 17 and x - y = 5, what is x?
If (2x + 3)/5 = 7, what is x?
If 4 - 3x > 13 and x is an integer, what is the greatest possible x?
If |x - 5| = 8, what is the sum of all real solutions?
An adult ticket costs 12 dollars and a child ticket costs 7 dollars. A group buys 9 tickets for 88 dollars. How many adult tickets are bought?
If 2x + 3y = 19 and x - y = 2, what is y?
For how many integers n does -2 <= 3n + 1 < 13 hold?
A rectangle has length 3 centimeters greater than twice its width and perimeter 42 centimeters. What is its width in centimeters?
A phone plan costs 18 dollars plus 0.04 dollars per message. What is the greatest integer number of messages that keeps the bill at or below 30 dollars?
A laboratory orders at least 40 sample tubes in boxes of 6. Each box costs 9 dollars. What is the least total cost in dollars?
Answer three reserved questions without hints. These cases are separate from lesson practice.
If 3u + v = 26 and u - v = 6, what is u?
A number is 4 less than three times another number. Their sum is 52. What is the smaller number?
If 7 - 2x <= -5 and x is an integer, what is the least possible value of x?
After a study gap, recover the inequality sign rule with a number-line example and solve a two-variable system by elimination. Verify both original equations rather than checking only the last line of working.
17. Complete an elimination, step 3
x = 5 and y = 4; both equations hold.
Checking only one equation would not verify a system solution.