SSAT Middle 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

3 practice questions for SSAT Middle Level

Q1 Medium

3 x - 5 < 17

For the inequality above, what is the greatest possible integer value of

x

A 5

B 6

C 7

D 8

E 9

Show Solution

First, solve the inequality for $x$ . Add 5 to both sides to get $3 x < 22$ . Next, divide both sides by 3 to get $x < \frac{22}{3}$ . Since $\frac{22}{3}$ is equal to $7 \frac{1}{3}$ , the inequality is $x < 7 \frac{1}{3}$ . The greatest integer less than $7 \frac{1}{3}$ is 7.

Answer: C

Q2 Medium

m

is a positive integer and

60 < m^{2} < 90

, which of the following is a possible value of

m + 3

A 10

B 11

C 13

D 14

E 15

Show Solution

First, determine the possible values for $m$ . The perfect squares between 60 and 90 are 64 (which is $8^{2}$ ) and 81 (which is $9^{2}$ ). Since $m$ is a positive integer, $m$ could be 8 or 9. If $m = 8$ , then $m + 3 = 8 + 3 = 11$ . If $m = 9$ , then $m + 3 = 9 + 3 = 12$ . Of these two possible values, only 11 is listed in the answer choices.

Answer: B

Q3 Medium

Sarah wants to read at least 60 pages of her book by the end of the week. She has already read 15 pages. If she plans to read exactly 7 pages each day, what is the minimum number of whole days she must read to reach her goal?

A 5

B 6

C 7

D 8

E 9

Show Solution

Let $d$ represent the number of days Sarah reads. The total number of pages she will have read is her starting amount plus the pages she reads over $d$ days, which is $15 + 7 d$ . She wants this total to be at least 60, so you can set up the inequality $15 + 7 d \geq 60$ . Subtract 15 from both sides to get $7 d \geq 45$ . Divide both sides by 7 to get $d \geq \frac{45}{7}$ . Since $\frac{45}{7}$ is approximately $6.42$ (or $6 \frac{3}{7}$ ), and Sarah must read for a whole number of days, the minimum number of days is the next integer up, which is 7.

Answer: C

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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