SSAT Upper Level

Inequalities

Q: Do I solve inequalities the exact same way as equations?

Almost! You can add, subtract, multiply, and divide to get `x` by itself, just like an equation. The only difference is the special rule: flip the inequality symbol if you multiply or divide by a negative number! 🔄

Q: What does a compound inequality like `2 < x < 5` mean?

It's a math sandwich! 🥪 It just means `x` is trapped between `2` and `5`. So `x` is bigger than `2`, but smaller than `5`.

Q: Will the SSAT ask me to graph these on a number line?

Yes! Remember that ` ` use an open circle (like a donut 🍩) because the number isn't included. `\leq` and `\geq` use a filled-in circle because the number IS included.

Q: Why does a fraction get smaller when I square it?

Think of it like taking a piece of a piece! If you have `\frac{1}{2}` of a candy bar, and you take `\frac{1}{2}` of THAT piece, you only have `\frac{1}{4}` of the whole bar left. 🍫

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

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Have you ever waited in line for an awesome roller coaster, only to see a sign that says, "You must be at least 48 inches tall to ride"? 🎢 That sign is an inequality! In math language, it means your height $h$ must be $h \geq 48$ . If you are exactly 48 inches, or 50 inches, or even 100 inches tall, you get to ride!

Unlike equations where things are perfectly balanced on a seesaw (like $x = 5$ ), inequalities are all about imbalance. One side is heavier, bigger, or smaller than the other. We use symbols like $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). 🍕 Think of the $<$ and $>$ symbols like a hungry alligator's mouth. The alligator always wants to eat the bigger slice of pizza! So, $3 < 8$ means the alligator is chomping the 8.

On the SSAT, solving inequalities is almost exactly like solving regular equations. You can add, subtract, multiply, and divide on both sides to find $x$ . But there is one sneaky ninja rule you MUST remember: if you multiply or divide by a negative number, the alligator flips around! Keep this in mind, and you'll crush these questions! 🥷✨

Key Formula

The Sneaky Ninja Rule: When you multiply or divide an inequality by a negative number, you MUST flip the inequality symbol! If

- a \cdot x < b

, then

x > \frac{b}{- a}

Practice Questions

5 practice questions for SSAT Upper Level

Q1 Hard

5 more than one-third of a number is less than 12.

Which of the following inequalities is equivalent to the statement above?

\frac{1}{3} x + 5 < 12

\frac{1}{3} x + 5 > 12

\frac{1}{3} (x + 5) < 12

5 x + \frac{1}{3} < 12

\frac{1}{3} x - 5 < 12

Show Solution

Let $x$ represent the unknown number. 'One-third of a number' is written as $\frac{1}{3} x$ . '5 more than' this quantity means we add 5, giving $\frac{1}{3} x + 5$ . Finally, 'is less than 12' translates to $< 12$ . Combining these components gives the inequality $\frac{1}{3} x + 5 < 12$ .

Answer: A

Q2 Hard

\frac{3}{4} - x < \frac{1}{2}

, which of the following could be a value of

x

\frac{1}{8}

\frac{1}{5}

\frac{1}{6}

\frac{2}{9}

\frac{1}{3}

Show Solution

First, solve the inequality for $x$ . Subtract $\frac{3}{4}$ from both sides to get $- x < \frac{1}{2} - \frac{3}{4}$ . Find a common denominator to subtract the fractions: $\frac{1}{2} = \frac{2}{4}$ . So, $- x < \frac{2}{4} - \frac{3}{4}$ , which simplifies to $- x < - \frac{1}{4}$ . Next, multiply or divide both sides by -1. Remember to flip the inequality sign when multiplying or dividing by a negative number! This gives $x > \frac{1}{4}$ . Since $\frac{1}{4} = 0.25$ , we need to find the answer choice that is greater than 0.25. Converting the choices to decimals or common denominators: $\frac{1}{3}$ is approximately 0.333, which is greater than 0.25. Therefore, $\frac{1}{3}$ is a possible value for $x$ .

Answer: E

Q3 Hard

A certain number

y

is at least -4 and strictly less than 7. Which of the following intervals describes all possible values of

y

- 4 < y < 7

- 4 \leq y \leq 7

- 4 \leq y < 7

- 4 < y \leq 7

y \leq - 4

y > 7

Show Solution

The phrase 'at least -4' means that $y$ can be -4 or any number greater than -4, which is written as $y \geq - 4$ (or $- 4 \leq y$ ). The phrase 'strictly less than 7' means $y < 7$ . Combining these two conditions into a single compound inequality yields $- 4 \leq y < 7$ .

Answer: C

Q4 Hard

The difference between 5 times a number

p

and half of another number

q

is at most 20.

The statement above can be represented by which of the following inequalities?

5 p - \frac{1}{2} q < 20

5 p - \frac{1}{2} q \leq 20

\frac{1}{2} q - 5 p \leq 20

5 p - \frac{1}{2} q \geq 20

5 p + \frac{1}{2} q \leq 20

Show Solution

'5 times a number $p$ ' is written as $5 p$ . 'Half of another number $q$ ' is written as $\frac{1}{2} q$ . The 'difference' between them indicates subtraction, so we write $5 p - \frac{1}{2} q$ . Finally, the phrase 'at most 20' means the quantity must be less than or equal to 20, which translates to $\leq 20$ . Combining these parts gives the inequality $5 p - \frac{1}{2} q \leq 20$ .

Answer: B

Q5 Hard

Which of the following describes all possible values of

x

- 2 x + 7 \geq 15

x \geq - 4

x \leq - 4

x \geq 4

x \leq 4

x \leq - 11

Show Solution

To solve the inequality $- 2 x + 7 \geq 15$ for $x$ , first subtract 7 from both sides to isolate the term with $x$ : $- 2 x \geq 8$ . Next, divide both sides by -2. Whenever you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality symbol. This gives $x \leq - 4$ .

Answer: B

Tips & Strategies

Pick a number! When a question gives you a range like $0 < x < 3$ , pick an easy number in that range (like $1$ or $2$ ) and plug it into the answer choices. It turns scary algebra into simple arithmetic! 🧠
Remember the fraction trap! Normally, multiplying a number makes it bigger (like $2^{3} = 8$ ). But if a number is a fraction between $0$ and $1$ (like $\frac{1}{2}$ ), multiplying it actually makes it smaller! 📉

Common Mistakes

Watch out for the negative flip! The #1 mistake students make is forgetting to flip the $<$ or $>$ sign when multiplying or dividing by a negative number. Always double-check your signs!
Don't forget that inequalities have many answers. If $x < 5$ , $x$ isn't just $4$ . It could be $4$ , $3$ , $0$ , $- 100$ , or even $4 \frac{1}{2}$ !

Frequently Asked Questions

Do I solve inequalities the exact same way as equations?

Almost! You can add, subtract, multiply, and divide to get $x$ by itself, just like an equation. The only difference is the special rule: flip the inequality symbol if you multiply or divide by a negative number! 🔄

What does a compound inequality like

2 < x < 5

mean?

It's a math sandwich! 🥪 It just means $x$ is trapped between $2$ and $5$ . So $x$ is bigger than $2$ , but smaller than $5$ .

Will the SSAT ask me to graph these on a number line?

Yes! Remember that $<$ and $>$ use an open circle (like a donut 🍩) because the number isn't included. $\leq$ and $\geq$ use a filled-in circle because the number IS included.

Why does a fraction get smaller when I square it?

Think of it like taking a piece of a piece! If you have $\frac{1}{2}$ of a candy bar, and you take $\frac{1}{2}$ of THAT piece, you only have $\frac{1}{4}$ of the whole bar left. 🍫

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