Solving Multi-Step & Compound Inequalities

Multi-step inequalities share many characteristics with multi-step equations, but solving them requires special awareness of inequality properties and algebraic manipulation. The solution for compound inequalities must consider all the possible values, and represent them using interval notation or a number line to accurately demonstrate the range that satisfies the initial inequality. Successfully navigating these topics helps students develop a solid base in algebra and understand the subtle, yet significant, distinctions between equations and inequalities.

Unlocking the Power of Inequalities: It’s Not Just About Being ‘Equal’!

Okay, math enthusiasts (and those who are reluctantly along for the ride!), let’s talk about something that goes beyond the typical equals sign. We’re diving into the world of inequalities! Now, I know what you might be thinking: “Ugh, more math?!” But trust me, this stuff is actually super useful, way beyond just torturing you in algebra class.

Think of it this way: equations are like saying you have to have exactly $10.00. Inequalities? They’re more like saying you need at least $10.00 to buy that super-cool widget you’ve been eyeing. It’s about options, flexibility, and dealing with the fact that life isn’t always equal! Inequalities give us a range of possible solutions instead of just one specific answer.

So, what exactly are inequalities? Well, instead of the trusty equals sign (=), we use symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), and even ≠ (not equal to). These symbols tell us that two things aren’t necessarily the same, but there’s still a relationship between them. This is where the fun (yes, I said fun!) begins. Over the course of this blog post, we’ll be looking at the following types of inequalities: multi-step, compound, absolute value.

And why should you care? Because inequalities are everywhere in the real world! Budgeting? That’s an inequality – you can’t spend more than you have (hopefully!). Driving? Speed limits are inequalities – you can’t go over a certain speed (unless you want a ticket!). Inequalities help us set limits, define boundaries, and make smart decisions in all sorts of situations. So, buckle up, because we’re about to unlock the power of inequalities!

Diving Deep: The Anatomy of an Inequality

Alright, let’s get down to brass tacks. Before you can wrestle those multi-step inequalities into submission, you gotta know what you’re actually looking at. Think of it like trying to assemble furniture without knowing the difference between a screw and a bolt – disaster! So, we’re going to break down the core components. Consider this your inequality anatomy lesson!

Variables: The Mystery Guests

First up, we have variables. Think of them as the mystery guests at your math party. They’re usually hanging out as a letter – x, y, z (but honestly, they could be any letter!). These represent unknown quantities, the very things we’re trying to solve for. Without them, you just have a plain old numerical statement. Variables turn your statement into an algebraic inequality, which is a fancy way of saying “something we can actually work with”.

Constants: The Reliable Numbers

Next, say hello to constants. These are your steadfast, reliable numerical values. They don’t change, they don’t waver, they’re just… there. Like the number 5, or -3.14 (yeah, even pi can be a constant!). Constants help define the boundaries of your solution. They’re the fixed points that influence the range of possibilities for your variable. Basically, they tell you what not to go beyond.

Coefficients: The Variable’s Bodyguards

Now, introducing coefficients. These are the numerical sidekicks glued to your variables. They’re the numbers multiplying the variable (like that ‘2’ in ‘2x’). They tell you how much each variable contributes to the overall expression. Mess with the coefficient, and you directly mess with the value of the variable.

Terms: The Individual Players

We have arrived at the terms! In an inequality, terms are those individual building blocks separated by addition or subtraction signs. So, in the inequality ‘3x + 2 > 7,’ ‘3x,’ ‘2,’ and ‘7’ are all terms. Think of them as the individual players on a sports team, each contributing to the overall score (or, in this case, the inequality).

Expressions: The Two Sides of the Story

And lastly, but definitely not least, there is expressions. An expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, etc.). In an inequality, you have two expressions separated by one of those inequality symbols (<, >, ≤, ≥). So, they form the sides of the inequality, like two teams competing for who has the bigger score.

Understanding these building blocks is absolutely essential for manipulating and solving inequalities. Without it, you’re basically trying to build a house with only half the materials – it ain’t gonna work! Once you get these down, you’re well on your way to becoming an inequality maestro!

Addition Property: It’s All About Balance (and Adding)!

Imagine an inequality as a perfectly balanced scale. On one side, you’ve got your expression, and on the other, you have another expression. The inequality symbol (<, >, ≤, ≥) tells you which side is heavier or if they’re equal in weight. Now, if you add the same amount of weight to both sides of the scale, what happens? Nothing! The scale remains balanced (or unbalanced in the same way it was before). That’s precisely what the addition property of inequality states: Adding the same value to both sides of an inequality doesn’t change the inequality.

Example:

Let’s say you have the inequality x – 3 > 5. To isolate x, you add 3 to both sides:

x – 3 + 3 > 5 + 3

This simplifies to x > 8. The solution set remains unaffected. The key takeaway is that whatever you add to one side, you must add the same to the other.

Subtraction Property: The Flip Side of Addition

Just as adding the same value to both sides keeps the inequality balanced, so does subtracting the same value. Think of it as removing equal weight from both sides of our balanced scale. The relationship between the two sides remains the same.

The Subtraction Property dictates that subtracting the same value from both sides of an inequality preserves the inequality. It mirrors the addition property perfectly.

Example:

Consider the inequality y + 7 ≤ 12. To solve for y, we subtract 7 from both sides:

y + 7 – 7 ≤ 12 – 7

This simplifies to y ≤ 5. Again, the solution set is consistent because we maintained balance by subtracting evenly.

Multiplication Property: Proceed with Caution!

Here’s where things get a little spicy. Multiplication is mostly straightforward, except when you’re multiplying by a negative number.

• Multiplying by a Positive Number: If you multiply both sides of an inequality by a positive number, the inequality sign stays the same. It’s like scaling up both sides of our balance proportionally; the relationship is still maintained.

• Multiplying by a Negative Number: HERE’S THE BIG WARNING! When you multiply both sides of an inequality by a negative number, you must FLIP THE INEQUALITY SIGN.

WHY? Multiplying by a negative number essentially reverses the direction of the number line. What was once greater becomes less, and vice versa.

Examples:

1. Positive Multiplication: If 2z < 10, multiplying both sides by 3 gives 6z < 30. The ‘<‘ sign remains.
2. Negative Multiplication: If -z > 4, multiplying both sides by -1 (to solve for z) gives z < -4. Notice the ‘>’ sign flipped to ‘<‘! If you don’t flip this, the sign will be incorrect.

Division Property: The Same Rules Apply

Division is just multiplication in reverse, which means the same rules apply. Dividing by a positive number maintains the inequality’s direction, while dividing by a negative number requires you to flip the inequality sign.

Dividing by a Positive Number: Dividing both sides of an inequality by a positive number maintains the integrity of the inequality.

Dividing by a Negative Number: Just like multiplication, dividing by a negative requires flipping the inequality sign! This is non-negotiable.

Examples:

1. Positive Division: If 4w ≥ 16, dividing both sides by 4 gives w ≥ 4. The ‘≥’ sign remains.
2. Negative Division: If -2w ≤ 8, dividing both sides by -2 gives w ≥ -4. Again, the ‘≤’ sign flipped to ‘≥’.

Transitive Property: The Chain Reaction

The transitive property is about creating a chain of inequalities. This property states:

If a < b and b < c, then a < c.

It can also be applied if a > b and b > c, then a > c.

In simpler terms, If one value is greater than another, and the second value is greater than the third, then the first value is greater than the third.

Example:

Suppose you know that x < y and y < 5. The transitive property lets you conclude that x < 5. It’s a handy shortcut for comparing multiple values at once.

Step-by-Step Guide to Solving Multi-Step Inequalities

Alright, buckle up, because we’re about to dive into the world of multi-step inequalities. Think of them as equations’ slightly more adventurous cousins. They’re not just looking for one right answer; they’re cool with a whole range of possibilities!

So, how do we tame these mathematical beasts? It’s all about breaking them down into smaller, manageable steps. We’re talking about simplification, variable isolation (no one likes feeling alone!), and a crucial sign-flipping maneuver when things get a little negative. Don’t worry, we’ll walk through it together, step-by-step, so you can tackle these problems with confidence.

Multi-Step Process

Let’s unpack this step-by-step.

Simplifying

First things first: Simplifying! Imagine your inequality is a messy room. The first step is to tidy up. That means combining all the “like terms” on each side. Like terms are just terms that have the same variable raised to the same power (or no variable at all – just constants!).

• Example: Let’s say you have 3x + 5 - x + 2. You can combine the 3x and -x to get 2x, and the 5 and 2 to get 7. So, the simplified expression is 2x + 7. Voila!

Isolating the Variable

Next up: Isolating the Variable. The goal here is to get your variable (usually x) all by itself on one side of the inequality. Think of it as giving x some personal space. To do this, we use something called “inverse operations.” This is where we start unwrapping the operations from the variable to leave just the variable alone.

• Inverse operations are simply operations that “undo” each other. Addition and subtraction are inverses, and multiplication and division are inverses. So, if something is being added to the variable, we subtract it from both sides. If something is being multiplied by the variable, we divide both sides.

  For example, if you have 2x + 3 > 7, you would first subtract 3 from both sides to get 2x > 4. Then, you would divide both sides by 2 to get x > 2.

Reversing the Inequality Sign

Okay, pay close attention because this is crucial. Reversing the Inequality Sign. This happens when you multiply or divide both sides of the inequality by a negative number. Why? Well, multiplying or dividing by a negative number flips the number line! So, to keep the inequality true, you have to flip the sign.

• Important Reminder: When you multiply or divide by a negative number, you must reverse the direction of the inequality sign. If it was “>” it becomes “<“. And vice versa. If it was “≥” it becomes “≤”. And vice versa.
• Example: Let’s say you have -3x < 9. If you divide both sides by -3, you must flip the inequality sign, so you get x > -3.

Solution Set

Now, let’s talk about the Solution Set. This is simply the range of all the possible values that make the inequality true. Think of it as a VIP list for numbers that can hang out in your inequality. Instead of a single solution, it is the set of all possible solutions.

For example, if you solve an inequality and get x > 5, the solution set is all the numbers greater than 5.

Equivalent Inequalities

Finally, let’s explore Equivalent Inequalities. Just like synonyms in English, different-looking inequalities can have the same meaning. They might look different, but they have the same solution set. These different expressions, but the same solution set, are referred to as equivalent inequalities.

• Example: x + 2 > 5 and x > 3 are equivalent inequalities because they both mean “x is greater than 3.”

Visualizing Solutions: Representing Inequalities Graphically

Okay, so you’ve wrestled with the algebraic side of inequalities, and you’re now staring at a solution like “x > 5.” But what does that really mean? It’s time to unleash the power of visualization! Think of representing inequalities graphically as translating from Math-speak to Picture-speak. Instead of just symbols, we’re going to paint the solution!

Number Line Representation: Your Inequality’s Portrait

The number line is our canvas. It’s where we bring our solutions to life!

• The Basics: Draw a horizontal line with arrows on both ends (to show it goes on forever!). Mark zero somewhere in the middle and then mark numbers to the left and right with equal spacing.

• Open vs. Closed Circles: This is crucial.

  • If your inequality is strict (using < or >), you use an open circle at the endpoint. An open circle means we approach the number but don’t include it in the solution. Imagine it like a velvet rope; you can admire the celebrity, but you can’t join them.
  • If your inequality includes “or equal to” (using ≤ or ≥), you use a closed circle. A closed circle is like a VIP pass; it includes the number.

• Shading: This shows all the numbers that make the inequality true. Shade to the left or right of your circle, depending on the inequality:

  • For x > a, shade to the right (all numbers greater than a).
  • For x < a, shade to the left (all numbers less than a).
  • For x ≥ a, shade to the right (all numbers greater than or equal to a).
  • For x ≤ a, shade to the left (all numbers less than or equal to a).

  Example:

  • To represent x < 3, draw an open circle at 3 and shade left.
  • To represent x ≥ -2, draw a closed circle at -2 and shade right.

Interval Notation: The Shorthand for Solutions

Imagine you’re texting your friend the solution to an inequality. Ain’t nobody got time to write out “all numbers greater than or equal to 7”! That’s where interval notation swoops in to save the day.

• Parentheses vs. Brackets: These are your key tools!

  • Parentheses (()): Use these for endpoints that aren’t included (like with < and >). Think of it as saying, “We’re getting close, but not touching!”
  • Brackets ([]): Use these for endpoints that are included (like with ≤ and ≥). This is a solid “Yep, we’re IN there!”

• Infinity: Because solutions can go on forever, we use the infinity symbol (∞). And because infinity is just a concept, and we can never actually reach it, it always gets a parenthesis!
  Note: Negative infinity will use the *negative sign.

Examples:

*   `x > 5` becomes `(5, ∞)` - All numbers *greater than* 5, going on to infinity.
*   `x ≤ -2` becomes `(-∞, -2]` - All numbers *less than or equal to* -2, going on to negative infinity.
*   `-1 < x ≤ 4` becomes `(-1, 4]` - All numbers *between* -1 (not included) and 4 (included).

Set-Builder Notation: Solutions with Style

Set-builder notation is like describing your solution in a fancy, mathematical way. It’s the poetry of inequalities!

• The Structure: It looks like this: {x | condition}

  • {}: This is the set. Think of it like a bag holding all the solutions.
  • x: This means “all the values of x…”
  • |: This is read as “such that” or “where.” It’s the dividing line between what we’re looking for and the conditions it must meet.
  • condition: This is the actual inequality that defines the solution.

Examples:

*   `x > 3` becomes `{x | x > 3}` - "The set of all x, *such that* x is greater than 3."
*   `x ≤ 7` becomes `{x | x ≤ 7}` - "The set of all x, *such that* x is less than or equal to 7."
*   `-2 ≤ x < 5` becomes `{x | -2 ≤ x < 5}` - "The set of all x, *such that* x is greater than or equal to -2 AND less than 5."

Visualizing inequalities isn’t just about drawing lines and circles. It’s about understanding what the solution truly means. With these tools, those abstract solutions become concrete, and you can see the range of possibilities.

Solving Compound Inequalities: Navigating the “Ands” and “Ors” of Math!

Alright, buckle up, math adventurers! We’re diving into the wild world of compound inequalities. Think of them as the VIP sections of the inequality club – they’re a little exclusive, requiring extra finesse to crack. Essentially, compound inequalities are formed when you take two or more inequalities and link them together using, you guessed it, “and” or “or“. It’s like math decided to play matchmaker, but with a twist!

“And” Inequalities: Where Solutions Intersect

Picture this: you’re throwing a party, but there are two rules for entry. You must be over 21 and bring a dish to share. Only those who meet both conditions get in! That’s the essence of “and” inequalities. They demand that the solution satisfies both inequalities simultaneously. We are seeking an intersection between the individual solution sets, the values that work for both inequalities.

Example: Let’s say we have:

1. x > 3
2. x < 7

To solve this “and” inequality, we need to find the values of x that are both greater than 3 and less than 7.

Graphically, we’d represent this on a number line with an open circle at 3 (since x is strictly greater than 3) and an open circle at 7 (since x is strictly less than 7), and shade the area between them.

In interval notation, the solution is (3, 7). It means that the result must be greater than 3 and less than 7.

“Or” Inequalities: Embrace the Union

Now, imagine a different kind of party: you can get in if you’re wearing a costume or if you’re a friend of the host. You only need to meet one of the criteria! That’s an “or” inequality. The solution needs to satisfy at least one of the inequalities. Here, we look for the union of the individual solution sets, values that work for either inequality.

Example: Consider these inequalities:

1. x < -2
2. x > 5

To solve this “or” inequality, we seek values of x that are either less than -2 or greater than 5.

On a number line, we’d have an open circle at -2 (since x is strictly less than -2), shading to the left, and an open circle at 5 (since x is strictly greater than 5), shading to the right. There is a gap between the two sections.

In interval notation, the solution is (-∞, -2) ∪ (5, ∞). The symbol ∪ means union, it means the range of values that satisfies either inequality.

Step-by-Step Solving of Compound Inequalities

Alright, now let’s roll up our sleeves and get practical! Here’s a general approach for tackling these compound beasts:

1. Isolate: Solve each inequality separately to isolate the variable. Remember those properties of inequalities!
2. Graph: Represent each solution set on a number line. This is super helpful for visualizing the “and” or “or” relationship.
3. Identify the Intersection or Union:
   • For “and” inequalities, find where the two solution sets overlap. That overlapping region is your final solution.
   • For “or” inequalities, combine both solution sets. Everything that’s shaded on either number line is part of your solution.
4. Express the Solution: Write your solution using interval notation or set-builder notation (we will cover that in the next outline section of the main article).

Example Time!

Let’s tackle this compound inequality:

-3 < 2x + 1 ≤ 7

This is secretly an “and” inequality in disguise! It’s saying that 2x + 1 must be greater than -3 and less than or equal to 7.

1. Separate and Isolate:

   • -3 < 2x + 1 => -4 < 2x => -2 < x
   • 2x + 1 ≤ 7 => 2x ≤ 6 => x ≤ 3
2. Combined Inequality: −2 < x ≤ 3.
3. Graph it: Draw a number line with an open circle at -2 (since x is strictly greater than -2) and a closed circle at 3 (since x is less than or equal to 3), and shade the area in between.
4. Interval Notation: Write the solution in interval notation: (-2, 3].

See? Not so scary once you break it down! Remember, the key is to treat each inequality individually and then think about how the “and” or “or” connects them. Happy solving!

Confronting Absolute Value Inequalities

Alright, buckle up buttercups, because we’re about to dive into the slightly weird world of absolute value inequalities. If regular inequalities are like saying, “I need at least 5 cookies,” absolute value inequalities are like saying, “I need to be within 2 cookies of having 5 cookies!” See? A bit more nuance, but totally doable.

What’s the deal with the absolute value sign? Well, think of it as a distance from zero. So, |x| < 3 means x is less than 3 units away from zero. That could be 2, 1, 0, -1, or -2! This is why we get two cases. This also the unique characteristics of absolute value inequalities.

Solving Absolute Value Inequalities: A Two-for-One Special

The key to cracking these bad boys is recognizing that the absolute value opens the door to two potential scenarios: a positive one and a negative one.

Here’s the process:

1. Isolate the Absolute Value: Get that absolute value expression all by itself on one side of the inequality. No clinging constants or coefficients allowed!

2. Split into Two Cases: This is where the magic happens. You’re going to create two separate inequalities.

   • Case 1: The Positive Case: Simply remove the absolute value bars and keep the inequality sign the same. Easy peasy.
   • Case 2: The Negative Case: Remove the absolute value bars, flip the inequality sign, and change the sign of the value on the other side of the inequality. This is crucial!

3. Solve Each Inequality: Now you’ve got two ordinary inequalities. Solve each one separately.

4. Combine the Solutions: Depending on the original inequality, you’ll either find the intersection (“and” scenario) or the union (“or” scenario) of the two solution sets.

   • If the original inequality was |x| < a (less than), you’re looking for the intersection – the values that satisfy both inequalities. This is an “and” situation.
   • If the original inequality was |x| > a (greater than), you’re looking for the union – the values that satisfy either inequality. This is an “or” situation.

Step-by-Step Examples

Let’s get our hands dirty with an example or two!

Example 1: |x – 2| < 3

1. The absolute value is already isolated!

2. Split into two cases:

   • Case 1: x – 2 < 3
   • Case 2: x – 2 > -3 (Notice the flipped sign!)

3. Solve each inequality:

   • Case 1: x < 5
   • Case 2: x > -1

4. Combine the solutions: Since the original inequality was “less than,” we need the intersection. The solution is -1 < x < 5. (All values of x greater than -1 and less than 5)

Example 2: |2x + 1| ≥ 5

1. The absolute value is already isolated!

2. Split into two cases:

   • Case 1: 2x + 1 ≥ 5
   • Case 2: 2x + 1 ≤ -5 (Again, flipped sign!)

3. Solve each inequality:

   • Case 1: 2x ≥ 4 => x ≥ 2
   • Case 2: 2x ≤ -6 => x ≤ -3

4. Combine the solutions: Since the original inequality was “greater than or equal to”, we need the union. The solution is x ≤ -3 or x ≥ 2. (All values of x less than or equal to -3 or greater than or equal to 2).

See? Not so scary after all! Remember to take it one step at a time, watch out for that sign flip, and you’ll be conquering absolute value inequalities like a pro.

Special Considerations: Extraneous Solutions

Okay, so you’ve become a solving inequalities wizard, huh? You can handle multi-step problems, graph solutions like a pro, and even wrangle those tricky compound inequalities. But hold on to your hats, folks, because there’s a sneaky little gremlin hiding in the world of inequalities: extraneous solutions.

Think of it like this: You’re baking cookies (because who isn’t thinking about cookies?). You follow the recipe perfectly, but somehow, one batch comes out tasting…off. It looks like a cookie, it smells like a cookie, but it definitely doesn’t taste like a cookie. That, my friends, is an extraneous solution – a solution that pops out during your calculations, struts around like it belongs, but ultimately doesn’t work when you plug it back into the original problem. Bummer.

So, what are these “extraneous solutions,” and why do they rear their ugly heads? Well, an extraneous solution is essentially a value that seems like it solves the inequality after you’ve done all your algebraic maneuvering, but it fails the acid test: when you substitute it back into the original inequality, it doesn’t hold true. These sneaky solutions often pop up when we perform operations that aren’t reversible or that can introduce new solutions that weren’t there before.

A common culprit for these unexpected imposters is working with absolute value inequalities or inequalities involving radicals. For example, when we are squaring both sides of an inequality to get rid of a pesky square root, or taking even roots we can inadvertently create a new inequality that has solutions that don’t work for the original.

Imagine you’re trying to solve an absolute value problem: |x| = -2. You might go through the steps, split it into two cases, and think you’ve found a solution. But wait! Absolute values are always non-negative. No number, when put inside absolute value bars, will ever equal a negative number. Therefore, any solution you might find along the way would be extraneous.

The key to outsmarting these sneaky solutions is simple: always, always, always check your answers. Treat the original inequality like the bouncer at a VIP party. Before you let any solution through, make sure it actually belongs there. Plug it back in, do the math, and see if it satisfies the original inequality. If it doesn’t, politely escort it out – it’s extraneous!

Consider the inequality √(x+2) > x . By solving this, we square both sides: x+2 > x^2. Rearranging we get x^2 – x – 2 < 0 or (x-2)(x+1) < 0. Which means -1< x < 2. But now we MUST check solutions.

• Let’s test x=0, √(0+2) > 0, which simplifies to √2 > 0. This is valid.
• Let’s test x=-2, √(-2+2) > -2, which simplifies to 0 > -2. This is valid.
• Let’s test x=3, √(3+2) > 3, which simplifies to √5 > 3, however, √5 is approximately 2.23, so 2.23 > 3 is NOT VALID.

So, always double-check your answers by plugging it back into the original inequality to make sure that the inequality stays true and see if the value is an actual solution. If not, it’s an extraneous solution. This simple step can save you from a lot of headaches and ensure you get the correct answer every time. Happy solving!

Real-World Applications: Inequalities in Action

Alright, let’s ditch the abstract and dive into the real world, where inequalities aren’t just symbols on paper, but the unsung heroes of everyday decisions! Forget those dusty textbooks; we’re about to see how these mathematical marvels are secretly running the show.

Budgeting: The “Can I Afford This?” Inequality

Ever stared longingly at that shiny new gadget or those designer shoes? Well, your brain was probably wrestling with an inequality! Budgeting is all about setting limits, right? It’s the ultimate balancing act of income versus expenses. Let’s say you earn $500 a month and need to save at least $100. You also have fixed expenses of $200.

Your inequality looks like this:

Expenses + Savings + Spending ≤ Income
$200 + $100 + Spending ≤ $500

Solving this tells you exactly how much wiggle room you have for fun stuff!

Optimization Problems: The Art of Squeezing Every Last Drop

Imagine you’re running a lemonade stand (a classic!). You want to make the most money possible, but you have constraints. Maybe you only have 20 lemons and 15 cups of sugar. Each glass of lemonade needs one lemon and one cup of sugar, and you sell each glass for 50 cents.

How do you maximize profit?

Constraints:
Lemons used ≤ 20
Cups of Sugar used ≤ 15

Profit per glass: 50 cents.

The maximum you can profit then is $7.50

Setting Constraints: When Rules Keep Us Safe

Think about speed limits. They’re not just arbitrary numbers; they’re constraints designed to keep us safe. Let’s say the speed limit is 65 mph. If “s” is your speed, then:

s ≤ 65

Going even slightly over can have serious consequences. Other examples? Temperature ranges for cooking food properly (to avoid food poisoning) or the maximum weight allowed on a bridge. Inequalities are all around us, ensuring things stay within safe and optimal boundaries.

So, there you have it! Inequalities aren’t just abstract math concepts; they are powerful tools that help us navigate the real world, from managing our budgets to optimizing our lemonade stands.

Alright, you’ve now got the tools to tackle those multi-step inequalities! It might seem tricky at first, but with a little practice, you’ll be solving them like a pro in no time. Good luck, and happy problem-solving!