Solve the inequality (38 > 4x + 3 > 7 - 3x)
Ever stared at a string of numbers and symbols and thought, “What on earth does this even mean?” You’re not alone. Double‑inequalities—those “A < B < C” statements—show up in textbooks, test prep, and even everyday budgeting problems, yet many people never learn how to untangle them. The short version is: treat each part like a mini‑equation, keep the direction of the inequality straight, and you’ll end up with a clean interval for x.
Below is a step‑by‑step walk‑through of the specific inequality
[
38 > 4x + 3 > 7 - 3x,
]
plus a deeper dive into why the method works, common slip‑ups, and a handful of practical tips you can reuse on any double‑inequality that pops up in a math class or on a standardized test.
What Is This Inequality?
In plain English, the statement says that the expression (4x + 3) must be smaller than 38 and larger than the expression (7 - 3x) at the same time. Think of it as a sandwich: the middle filling (our (4x+3)) has to fit snugly between the top slice (38) and the bottom slice (7 – 3x).
Mathematically, a double‑inequality is just a compact way of writing two separate inequalities that share a common term. If you split it apart, you get:
- (38 > 4x + 3) (the “upper bound”)
- (4x + 3 > 7 - 3x) (the “lower bound”)
Both must hold true simultaneously. The goal is to find every x that satisfies both conditions.
Why It Matters
You might wonder why we bother with this kind of algebra. In practice, double‑inequalities pop up whenever you need a variable to stay within a safe range.
- Engineering: Tolerances on a component’s dimension are often expressed as “must be greater than X but less than Y.”
- Finance: A budget constraint could read “spending must stay below $38 and above $7 – 3x” (where x is a variable expense).
- Standardized tests: The SAT, ACT, and many college‑level exams love to throw a double‑inequality at you as a quick “do you understand the rules?” check.
If you can solve them cleanly, you avoid costly mistakes—like ordering a part that won’t fit or budgeting for a month you can’t actually afford That's the part that actually makes a difference..
How It Works
Step 1 – Separate the Two Inequalities
Write them out side by side:
38 > 4x + 3 (1)
4x + 3 > 7 - 3x (2)
Now you have two simpler problems to tackle Not complicated — just consistent..
Step 2 – Solve the Upper‑Bound Inequality (1)
[ 38 > 4x + 3 ]
Subtract 3 from both sides:
[ 35 > 4x ]
Divide by 4 (positive, so the direction stays the same):
[ \frac{35}{4} > x \quad\text{or}\quad x < 8.75. ]
So far, x must be less than 8.75 Simple, but easy to overlook. Nothing fancy..
Step 3 – Solve the Lower‑Bound Inequality (2)
[ 4x + 3 > 7 - 3x ]
First, get all the x terms on one side. Add (3x) to both sides:
[ 7x + 3 > 7 ]
Now subtract 3:
[ 7x > 4 ]
Divide by 7:
[ x > \frac{4}{7} \approx 0.571. ]
Now we have the second condition: x must be greater than 4⁄7.
Step 4 – Combine the Results
Both conditions must hold, so we intersect the two intervals:
[ \frac{4}{7} < x < \frac{35}{4}. ]
In decimal form:
[ 0.571\ldots < x < 8.75. ]
That’s the solution set. In interval notation it’s ((\frac{4}{7},,\frac{35}{4})) Easy to understand, harder to ignore..
Step 5 – Double‑Check with a Test Value
Pick a number inside the interval, say (x = 2).
- Upper bound: (4(2)+3 = 11 < 38) ✅
- Lower bound: (4(2)+3 = 11 > 7 - 3(2) = 1) ✅
Both are true, so the interval checks out. Try a number outside, like (x = 10). In real terms, the upper bound fails because (4(10)+3 = 43) is not less than 38. Good sanity check It's one of those things that adds up..
Common Mistakes / What Most People Get Wrong
-
Flipping the inequality sign when dividing by a negative number.
In our example we never divided by a negative, but if the lower‑bound step had been ( -2x > 5), you’d need to reverse the sign to get (x < -2.5). Forgetting that reversal flips the whole solution set. -
Treating the double‑inequality as a single “chunk.”
Some students try to move terms across both inequality symbols at once, ending up with nonsense like (38 - 4x + 3 > 7). The safe route is to split, solve, then recombine That's the part that actually makes a difference. Took long enough.. -
Missing the “and” requirement.
The solution isn’t the union of the two individual solutions; it’s the intersection. If you wrote (x > 4/7) or (x < 35/4), you’d mistakenly include every real number—clearly wrong Took long enough.. -
Rounding too early.
Converting fractions to decimals before you finish the algebra can introduce tiny errors that compound. Keep fractions until the final step, then decide whether you want a decimal Worth knowing.. -
Assuming the inequality signs are all “greater than.”
The original statement had a “greater than” on the left side and a “greater than” on the right side, but the middle term is sandwiched. Swapping them changes the whole problem Less friction, more output..
Practical Tips – What Actually Works
- Write each inequality on its own line. Visual separation prevents accidental sign flips.
- Isolate the variable early. Move constants away from the x term before you start dividing or multiplying.
- Keep track of the direction. Whenever you multiply or divide by a negative, write a quick note: “flip”.
- Use interval notation. It forces you to think about open vs. closed ends (here both are open because the original symbols are strict “>” and “<”).
- Test two numbers: one inside the interval, one just outside. It’s a cheap sanity check that catches sign errors.
- When in doubt, draw a number line. Mark the two critical points ((4/7) and (35/4)) and shade the region that satisfies both conditions.
Applying these habits will make any double‑inequality feel like a routine check rather than a brain‑twister.
FAQ
Q1: What if the inequality had “≥” or “≤” signs?
A: Treat them the same way, but remember that the interval becomes closed at that end. As an example, (38 \ge 4x+3 > 7-3x) would give (x \le 35/4) and (x > 4/7), so the solution is ((4/7,,35/4]).
Q2: Can I solve a triple‑inequality like (a < b < c < d) in one go?
A: Yes—break it into three separate inequalities, solve each, then intersect all three solution sets Not complicated — just consistent..
Q3: Does the order of the numbers matter?
A: Absolutely. Swapping the left‑most and right‑most numbers changes which side is the upper bound and which is the lower bound, flipping the whole solution That's the part that actually makes a difference..
Q4: How do I handle absolute values inside a double‑inequality?
A: Split the absolute value into its two cases (positive and negative) first, then treat each case as its own double‑inequality.
Q5: Is there a shortcut for linear double‑inequalities?
A: For linear expressions, you can sometimes combine them into a single inequality by subtracting the middle term from both outer terms, but only if you’re careful with signs. Most students find the “split‑and‑solve” method less error‑prone And it works..
That’s it. You’ve taken a string that at first looks like a cryptic code and turned it into a clear interval for x. Next time you see a double‑inequality, remember: split, solve, intersect, and double‑check. It’s a tiny process that saves a lot of headaches—whether you’re cramming for a test or just trying to keep your budget in the green. Happy solving!
The key to mastering double-inequalities is to break them down systematically and employ a few strategic habits. By isolating the variable early, keeping track of the direction of inequalities, and using interval notation, you can avoid common pitfalls like sign flips and misinterpretations of open versus closed intervals. Testing with specific numbers and using a number line can provide valuable checks to ensure the correctness of the solution.
Remember, the order of terms in a double-inequality is crucial, as it defines which values are upper and lower bounds. Here's the thing — handling absolute values or multiple inequalities requires splitting them into manageable parts and solving each before finding the common solution. While shortcuts exist for linear expressions, the split-and-solve method is generally safer and more reliable.
Easier said than done, but still worth knowing.
Pulling it all together, double-inequalities are not as intimidating as they seem. Now, with practice and these practical tips, you can approach them with confidence, knowing that each step—splitting, solving, intersecting, and verifying—leads you closer to the correct interval for your variable. Embrace the process, and you'll find that these inequalities are not just solvable, but also a great way to sharpen your algebraic skills.
