Substituting 5 for b confirms the inequality 5b + 2 ≥ 12 is true.

Inequalities show up all the time, even in everyday math—like when you want to know if a certain value makes a statement hold true. Here’s a clean, real-world way to think about one that pops up in the HSPT math section: what happens when you substitute a number for a variable? Let me walk you through a simple example and why it matters.

What substitution really means

Imagine you’re checking a statement that uses a variable, say b. The inequality is 5b + 2 ≥ 12. Substituting a number for b means you’re turning the algebra into a concrete number comparison. It’s not solving for b or finding all possible b that make the inequality true; it’s testing whether a single, specific value satisfies the inequality.

Think of it as like testing a condition in a video game. You pick a value for a variable (your character’s speed, say), apply the rule, and see if you meet the requirement (can you reach the next level). In math terms, you’re evaluating the truth of the inequality with that particular substitution.

Step-by-step with b = 5

Now, let’s substitute b = 5 into the inequality 5b + 2 ≥ 12.

• Start with the expression: 5b + 2 ≥ 12.

• Replace b with 5: 5(5) + 2 ≥ 12.

• Do the arithmetic: 25 + 2 ≥ 12.

• Simplify: 27 ≥ 12.

That statement is true. So, with b equal to 5, the inequality holds.

Important takeaway: substituting a value confirms whether that specific instance satisfies the condition. It’s a quick truth check, not a full solution set.

Why this kind of check shows up on the HSPT

Tests like the HSPT love to blend straightforward arithmetic with a touch of logic. Substitution checks a basic idea: does a given number cooperate with the rule? It teaches you to:

• Read the inequality carefully. The symbol ≥ means “is at least,” so you’re verifying that the left-hand side isn’t smaller than the right-hand side.

• Do the math cleanly. Multiply first, then add, then compare.

• Not confuse the act of substitution with solving for all possible values. Substitution asks about one specific value, while solving asks for the whole range of values that work.

A quick mental model

If you’re thinking about the inequality 5b + 2 ≥ 12, here’s a simple mental shortcut. You want the left side to be at least 12. If you plug in a positive number for b, the left side grows (because 5b grows as b grows). So, with larger b you’ll often make the inequality more likely to be true. Of course, that’s a general intuition; for a particular number you still do the exact arithmetic to be sure.

Common pitfalls to sidestep

Even when the math is simple, it’s easy to stumble. Here are a few missteps to watch out for:

• Forgetting to substitute correctly. It’s easy to drop a sign or misplace parentheses. Always replace every instance of b with the chosen value.

• Mixing up the order of operations. Multiply before you add. In our example, 5(5) is 25, not 5 + 5.

• Treating “≥” like “>.” Remember, 27 ≥ 12 is true, but if the numbers were 12 and 12, it would still be true.

• Thinking substitution means you’re solving for all possible b. That’s a broader task; here you’re just testing one specific value.

Transitioning from a single check to a broader skill

While this particular substitution exercise tests one value, you can scale the idea to more general problems. For instance, if you’re asked whether a specific b makes the inequality true for all potential b in a range, you’d compare the entire left-hand side’s behavior to the right-hand side across that range. The core thinking doesn’t change: plug in the value, compute, and compare.

Relatable analogies to keep the idea grounded

Imagine you’re checking a recipe’s yield. The inequality is like “at least 12 cookies will come out.” If you bake with b = 5 cups of flour, you estimate the dough’s amount and see if you reach that yield. If the bake clearly makes more than 12, you know the condition holds for that amount. Substitution is just testing one scenario in a kitchen-full of possibilities.

Practical tips for strengthening this skill

• Keep a little arithmetic routine in your head. Multiply, add, compare—fast, clean, and deliberate.

• Use a cheat-check: after you substitute, re-check the key operation. If you used 5(5) + 2, you know you did the multiplication first, then the addition.

• Practice with a few quick variations. Try b = 0, b = 1, and a larger value like b = 10. Notice how the left side responds to bigger b.

• Visualize the inequality. The left-hand side is a line in a graph: y = 5b + 2. The inequality y ≥ 12 asks, “Where is this line at or above the line y = 12?” Substituting b is like checking a specific x-value on that line.

Real-world relevance beyond test items

This kind of reasoning isn’t just for tests. It shows up when you read a budget, evaluate a discount offer, or even decide if a coupon code applies to your cart total. The habit is simple but powerful: test the rule with a concrete number, verify the math, and confirm what the statement means in plain terms.

A gentle reminder about tone and process

If you’re someone who likes a clear, step-by-step approach, this can feel almost soothing: substitute, calculate, compare, conclude. On the other hand, if you enjoy seeing patterns, you might notice that the larger the substituted value, the more likely the inequality becomes true for expressions like 5b + 2 ≥ 12. Either way, the essential move is the same: turn an abstract rule into a concrete yes or no.

Putting it all together

Here’s a compact recap:

• Substituting b = 5 into 5b + 2 ≥ 12 gives 27 ≥ 12.

• That statement is true, so the substitution confirms the inequality holds for that value.

• This exercise is a small but meaningful slice of mathematical reasoning that helps you with broader problem-solving skills.

Takeaway for readers

Whenever you see a variable in an inequality, testing a specific value by substitution is a quick truth check. It trains you to read the rule, perform the arithmetic correctly, and decide whether the statement stands for that value. It’s a simple, reliable tool—one you’ll reach for again and again, whether you’re tackling a handful of items on a test or sorting through a real-world puzzle that needs a crisp, logical answer.

If you’re curious, there are plenty of other friendly substitution scenarios you can try. Pick a few values for b, plug them into expressions like 5b + 2 ≥ 12, and watch how the truth of the statement changes as b moves. You’ll notice patterns, build confidence, and strengthen a core muscle of math thinking—the kind of reasoning that pays off in life as much as it does on a page.