How to Write Two-variable Inequalities Word Problems?

A Step-by-step Guide to Two-variable Inequalities Word Problems

Writing two-variable inequalities word problems involves several steps. Here’s a step-by-step guide:

Step 1: Identify the variables

Identify the two variables that will be used in the inequality. It’s essential to define what each variable represents to avoid any confusion in the problem.

Step 2: Determine the inequality symbol

Determine the inequality symbol that will be used in the problem. Common inequality symbols include “greater than,” “less than,” “greater than or equal to,” and “less than or equal to.” It’s important to select the correct symbol based on the problem’s context.

Step 3: Write the inequality

Write the inequality using the variables and the inequality symbol.

Step 4: Formulate the word problem

Write a word problem that uses the inequality. The word problem should be relevant to the inequality and include all necessary information, such as the values of the variables and any additional constraints.

Step 5: Check for consistency

Check the word problem to ensure that it’s consistent with the inequality. The problem should be solvable using the inequality, and the solution should satisfy the inequality. If there are any inconsistencies, revise the problem or inequality as necessary.

Writing Two-variable Inequalities Word Problems, Examples 1

John has \($500\) to spend on shirts and pants. Shirts cost \($20\) each, and pants cost \($30\) each. Write an inequality that represents the possible combinations of shirts and pants John can buy.

Solution:

Step 1: Identify the variables

Let \(x\) represent the number of shirts John can buy and \(y\) represent the number of pants he can buy.

Step 2: Determine the inequality symbol

Since John has a fixed amount of money, the inequality will involve an inequality symbol that limits the total cost of the shirts and pants. The inequality symbol in this case is \(“≤”\) (less than or equal to).

Step 3: Write the inequality

The inequality is: \(20x + 30y ≤ 500\)

Step 4: Formulate the word problem

What combinations of shirts and pants can John buy without spending more than \($500\)?

Step 5: Check for consistency

The word problem is consistent with the inequality.

Writing Two-variable Inequalities Word Problems, Examples 2

Samantha is planning a trip to the beach and wants to rent beach chairs and umbrellas. She has a budget of \($200\) and chairs cost \($10\) each, while umbrellas cost \($20\) each. Write an inequality that represents the possible combinations of chairs and umbrellas Samantha can rent.

Solution:

Step 1: Identify the variables

Let \(x\) represent the number of chairs Samantha can rent and \(y\) represent the number of umbrellas she can rent.

Step 2: Determine the inequality symbol

Since Samantha has a fixed amount of money, the inequality will involve an inequality symbol that limits the total cost of the chairs and umbrellas. The inequality symbol in this case is \(“≤”\) (less than or equal to).

Step 3: Write the inequality

The inequality is: \(10x + 20y ≤ 200\)

Step 4: Formulate the word problem

What combinations of chairs and umbrellas can Samantha rent without spending more than \($200\)?

Step 5: Check for consistency

The word problem is consistent with the inequality.

Key phrases and the inequality symbols they mean

A quick tip: “at most” and “at least” include the boundary value, so they use ≤ and ≥. “Less than” and “more than” do not include it.

Worked example with “at least” and a point check

Problem: Lena earns $12 an hour babysitting and $15 an hour tutoring. She wants to earn at least $180 this month. Write an inequality for the hours she could work.

1. Define the variables: x = hours babysitting, y = hours tutoring.
2. Write the earnings expression: 12x + 15y.
3. Pick the symbol: “at least” means ≥.
4. Write the inequality: 12x + 15y ≥ 180, with x ≥ 0 and y ≥ 0.
5. Check a point: 10 hours babysitting and 4 hours tutoring gives 12(10) + 15(4) = 120 + 60 = 180. Since 180 ≥ 180 is true, this plan works. Try 5 and 5: 60 + 75 = 135, which is less than 180, so that plan falls short.

Checking one point that should work and one that should not is a fast way to catch a backwards symbol.

Frequently Asked Questions

How do you write an inequality from a word problem?

Name the unknown with a variable, build an expression from the costs or rates in the story, then choose the symbol from the key phrase. For example, if a phone plan costs $15 plus $0.10 per text and you can spend no more than $25, the inequality is 15 + 0.10t ≤ 25. Always say what the variable represents.

How do you solve two-step inequality word problems?

Write the inequality, undo addition or subtraction, then undo multiplication or division. Example: a gym charges a $30 joining fee plus $12 per month, and you can spend at most $150. So 30 + 12m ≤ 150, then 12m ≤ 120, so m ≤ 10 months. If you divide by a negative number, flip the inequality symbol.

What words mean less than or greater than in inequalities?

Phrases like at most, no more than and maximum mean ≤. At least, no less than and minimum mean ≥. More than, exceeds and over mean >, while fewer than and under mean <. Watch for the word than changing the order: 5 less than x is x − 5, but 5 is less than x is 5 < x.

How do you write a two-variable inequality from a word problem?

Use one variable for each quantity and multiply each by its price or rate. Example: adult tickets cost $8 (x) and child tickets $5 (y), and the club must raise at least $400, so 8x + 5y ≥ 400. Test a point: 30 adults and 40 children give 240 + 200 = 440, which works. Add x ≥ 0 and y ≥ 0.

How do you tell a story about an inequality like 5 > y?

Decide what y counts, then describe a limit that keeps it under 5. For example, let y be the number of friends Maya can invite to a sleepover, and her parents say fewer than 5 guests. Then 5 > y, so y can be 0, 1, 2, 3 or 4. The constraint is the rule, and y is the amount it limits.