ISEE Middle Level

Inequalities

Q: How is an inequality different from a regular equation?

An equation has an equals sign and usually just one exact answer (like `x = 5`). An inequality has a greater-than or less-than sign, meaning there are infinite answers! Any number bigger than `5` would work for `x > 5`.

Q: What does it mean when the alligator mouth has a line under it?

That line means 'or equal to.' So, `\ge` means 'greater than OR equal to.' If the rule is `x \ge 3`, `x` can be `4`, `5`, or exactly `3`!

Solving and graphing linear inequalities and compound inequalities — uses < > ≤ ≥ instead of =

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Have you ever wanted to ride a giant, looping rollercoaster, but you saw a sign that said, 'You must be at least 48 inches tall to ride'? 🎢 That sign is actually a real-life math inequality!

In math, a regular equation is like hitting the exact center of a dartboard. It gives you one specific answer, like $x = 5$ . But an inequality is like a rule that gives you a whole range of winning choices! It tells you that a number can be greater than, less than, or between other numbers. If the rollercoaster rule is $h \geq 48$ (where $h$ is your height), you don't have to be exactly 48 inches tall. You can be 49 inches, 50 inches, or even 60 inches tall, and you still get to ride!

On the ISEE, you'll see inequalities using symbols like $<$ (less than) and $>$ (greater than). A great trick to remember is that the symbol is like a hungry alligator's mouth 🐊—it always opens up to eat the bigger number!

You will solve inequalities almost exactly the same way you solve regular equations. You can add, subtract, multiply, and divide to get $x$ by itself. The ISEE loves to test these in Quantitative Comparison questions, where you have to compare Column A and Column B. Are you ready to tame the math alligators? Let's go! 🚀

Key Formula

To solve an inequality, do the same thing to both sides! BUT, if you multiply or divide by a NEGATIVE number, you MUST flip the alligator's mouth! Example: If

- a x > b

, then

x < - \frac{b}{a}

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium

Solve the inequality

3 x - 7 < 8

x < \frac{1}{3}

x < 5

x > 5

x > \frac{1}{3}

Show Solution

To solve the inequality $3 x - 7 < 8$ , follow these steps:
1. Add 7 to both sides of the inequality: $3 x - 7 + 7 < 8 + 7$ which simplifies to $3 x < 15$ .
2. Divide both sides by 3: $rac{3x}{3} < rac{15}{3}$ which simplifies to $x < 5$ .

Thus, the solution is $x < 5$ .

Answer: B

Q2 Medium

A taxi charges a flat fee of $

2.50

plus $

1.50

per mile. Sarah wants to spend no more than $

10.00

on a taxi ride. Which inequality represents the maximum number of miles (

m

) Sarah can travel?

2.50 + 1.50 m < 10.00

2.50 + 1.50 m \leq 10.00

1.50 + 2.50 m \leq 10.00

2.50 m \leq 10.00

Show Solution

Let $m$ represent the number of miles Sarah travels.

The flat fee is $ $2.50$ .
The cost per mile is $ $1.50 m$ .
The total cost of the taxi ride is $2.50 + 1.50 m$ .
Sarah wants to spend "no more than" $ $10.00$ . This phrase translates to less than or equal to ( $\leq$ ).
Therefore, the inequality that represents the situation is $2.50 + 1.50 m \leq 10.00$ .

Answer: B

Q3 Medium

Solve the inequality

5 x + 12 \geq 2 x - 3

x \geq - 5

x \leq - 5

x \geq - 3

x \leq - 3

Show Solution

To solve the inequality $5 x + 12 \geq 2 x - 3$ , follow these steps:
1. Subtract $2 x$ from both sides of the inequality: $5 x - 2 x + 12 \geq 2 x - 2 x - 3$ which simplifies to $3 x + 12 \geq - 3$ .
2. Subtract $12$ from both sides: $3 x + 12 - 12 \geq - 3 - 12$ which simplifies to $3 x \geq - 15$ .
3. Divide both sides by $3$ : $rac{3x}{3} \ge rac{-15}{3}$ which simplifies to $x \geq - 5$ .

Thus, the solution is $x \geq - 5$ .

Answer: A

Q4 Medium

Emily has $

50

to spend at an amusement park. The entrance fee is $

15

, and each ride costs $

3.50

. What is the maximum number of rides (

r

) Emily can go on?

r < 10

r \leq 10

r < 11

r \leq 11

Show Solution

Let $r$ be the number of rides Emily can go on.

The entrance fee is $ $15$ .
The cost for $r$ rides is $3.50 r$ .
The total cost is $15 + 3.50 r$ .
Emily has $ $50$ to spend, so the total cost must be less than or equal to $ $50$ . This gives the inequality:
$15 + 3.50 r \leq 50$
Now, solve for $r$ :
1. Subtract $15$ from both sides: $3.50 r \leq 50 - 15$ which simplifies to $3.50 r \leq 35$ .
2. Divide both sides by $3.50$ : $r \le rac{35}{3.50}$ .
3. Calculate the division: $r \leq 10$ .

Since the number of rides must be a whole number, the maximum number of rides Emily can go on is $10$ .

Answer: B

Tips & Strategies

Plug in numbers! In Quantitative Comparison questions with inequalities, picking a simple number that fits the rule (like $x = 2$ when $x > 1$ ) makes the problem a breeze.
Check the boundaries. When dealing with a range like $- 3 < x < 2$ , see what happens if you plug in numbers really close to the edges, like $- 2.9$ or $1.9$ .
Remember that fractions are numbers too! If a question says $0 < x < 1$ , you can't pick a whole number. You have to pick a fraction like $\frac{1}{2}$ .

Common Mistakes

Watch out for the 'Negative Flip' trap! If you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality sign. $>$ becomes $<$ .
Don't forget that inequalities don't just include whole numbers. If $x < 2$ , $x$ could be $1$ or $0$ , but it could also be $1.5$ or $\frac{1}{2}$ !

Frequently Asked Questions

How is an inequality different from a regular equation?

An equation has an equals sign and usually just one exact answer (like $x = 5$ ). An inequality has a greater-than or less-than sign, meaning there are infinite answers! Any number bigger than $5$ would work for $x > 5$ .

Do I ever lose points for guessing on ISEE inequality questions?

Nope! The ISEE has no guessing penalty. If you're stuck on a tricky inequality, try your best to eliminate one or two wrong answers, and then guess. Never leave a bubble blank!

What does it mean when the alligator mouth has a line under it?

That line means 'or equal to.' So, $\geq$ means 'greater than OR equal to.' If the rule is $x \geq 3$ , $x$ can be $4$ , $5$ , or exactly $3$ !

Are Quantitative Comparison questions always on the ISEE?

Yes! You'll see them in the Quantitative Reasoning section if you are taking the Middle or Upper level ISEE. They always compare Column A and Column B. Picking your own numbers is your best strategy here!

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