Solve (4x+22)² > 0: Finding Values of x in a Perfect Square Inequality

Q: Why can't a squared expression be negative?

When you square any real number, you always get a positive result or zero. For example: $ 3^2 = 9 $ and $ (-3)^2 = 9 $. This is a fundamental property of squares!

Q: What does the notation x ≠ -5½ mean exactly?

This means all real numbers except -5.5. So x can be any value like -6, -2, 0, 10, etc., but it cannot equal -5.5 because that's where the function equals zero, not greater than zero.

Q: How do I solve 4x + 22 = 0 step by step?

Start with $ 4x + 22 = 0 $Subtract 22: $ 4x = -22 $Divide by 4: $ x = -\frac{22}{4} = -5.5 $

Q: Why isn't the answer 'all x' since squares are always positive?

Great thinking! Squares are always non-negative (≥ 0), but we need strictly greater than zero (> 0). At $ x = -5.5 $, the function equals exactly 0, which doesn't satisfy f(x) > 0.

Q: How can I verify this answer makes sense?

Test a few values! Try $ x = 0 $: $ (4(0)+22)^2 = 22^2 = 484 > 0 $ ✓Try $ x = -6 $: $ (4(-6)+22)^2 = (-2)^2 = 4 > 0 $ ✓Only at $ x = -5.5 $ does it equal 0.

Perfect Square Inequalities with Zero Values

Look at the function below:

$y=\left(4x+22\right)^2$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the function below:

$y=\left(4x+22\right)^2$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To solve this problem, we observe that the function given is $y = (4x + 22)^2$ .

Step 1: We set the expression inside the square equal to zero and solve for $x$ .
$4x + 22 = 0$

Step 2: Solve the equation above for $x$ :

4x + 22 = 0 \\ 4x = -22 \\ x = -\frac{22}{4} = -5.5

This calculation reveals that $x = -5.5$ is the only point where $(4x + 22)^2 = 0$ .

Step 3: Outside of this specific $x$ , the squared term $(4x+22)^2$ is positive for all other values of $x$ .

Therefore, the function is positive $(f(x) > 0)$ when $x \neq -5.5$ .

Thus, the solution to the problem is: $x \neq -5\frac{1}{2}$ .

Final Answer

$x\ne-5\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Square Property: Any squared expression is always non-negative
Technique: Set $4x + 22 = 0$ to find where expression equals zero
Check: At $x = -5.5$ , $(4(-5.5) + 22)^2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking squared expressions can be negative
Don't assume $(4x+22)^2$ can be less than zero = impossible result! Squared expressions are always ≥ 0, never negative. Always remember that any real number squared gives a positive result or zero.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

Why can't a squared expression be negative?

When you square any real number, you always get a positive result or zero. For example: $3^2 = 9$ and $(-3)^2 = 9$ . This is a fundamental property of squares!

What does the notation x ≠ -5½ mean exactly?

This means all real numbers except -5.5. So x can be any value like -6, -2, 0, 10, etc., but it cannot equal -5.5 because that's where the function equals zero, not greater than zero.

How do I solve 4x + 22 = 0 step by step?

Start with $4x + 22 = 0$
Subtract 22: $4x = -22$
Divide by 4: $x = -\frac{22}{4} = -5.5$

Why isn't the answer 'all x' since squares are always positive?

Great thinking! Squares are always non-negative (≥ 0), but we need strictly greater than zero (> 0). At $x = -5.5$ , the function equals exactly 0, which doesn't satisfy f(x) > 0.

How can I verify this answer makes sense?

Test a few values! Try $x = 0$ : $(4(0)+22)^2 = 22^2 = 484 > 0$ ✓
Try $x = -6$ : $(4(-6)+22)^2 = (-2)^2 = 4 > 0$ ✓
Only at $x = -5.5$ does it equal 0.

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Solve (4x+22)² > 0: Finding Values of x in a Perfect Square Inequality

Perfect Square Inequalities with Zero Values

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't a squared expression be negative?

What does the notation x ≠ -5½ mean exactly?

How do I solve 4x + 22 = 0 step by step?

Why isn't the answer 'all x' since squares are always positive?

How can I verify this answer makes sense?

🌟 Unlock Your Math Potential

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