Solve each equation separately and find which x is the largest.$ 3+x+17-56x=18x+44 $$ (x-2)(x+2)=0 $

There is no solution to the two equations

Solve and Compare: Tackle the Equations 3 + x + 17 - 56x = 18x + 44 and (x-2)(x+2) = 0

Q: Why does the second equation have two solutions?

The equation (x-2)(x+2) = 0 is a product of two factors. When a product equals zero, at least one factor must be zero. So either x-2 = 0 (giving x = 2) or x+2 = 0 (giving x = -2).

Q: How do I compare a fraction like -24/73 with whole numbers?

Convert the fraction to a decimal: -24/73 ≈ -0.33. Now you can easily compare: -2

Q: Can I expand (x-2)(x+2) instead of using zero product property?

Yes! Expanding gives x² - 4 = 0, then x² = 4, so x = ±2. Both methods work, but zero product property is usually faster for factored equations.

Q: What if I made an algebra mistake in the linear equation?

Always check your work! Substitute x = -24/73 back into the original equation 3 + x + 17 - 56x = 18x + 44. Both sides should give the same value.

Q: Why do we need to find the largest x from all equations?

The question asks us to compare solutions from different equations. We're not solving a system (where x must satisfy both), but finding which individual solution has the greatest value.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the largest X

00:03 First find X from section 1

00:15 Collect terms

00:30 Arrange the equation so that X is only on the left side

00:58 Collect terms

01:12 Isolate X

01:22 This is the solution for X in section 1

01:26 Now let's calculate X from section 2

01:39 Find the solutions that make the equation equal to zero

01:43 These are the possible solutions for X in section 2

01:50 We can see that in section 2 one of the possible solutions is the largest

01:54 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve each equation separately and find which x is the largest.

^{$3+x+17-56x=18x+44$}
$(x-2)(x+2)=0$

2

Step-by-step solution

Let's solve each equation step-by-step and find the largest $x$ .

Equation 1: $3 + x + 17 - 56x = 18x + 44$

This is a linear equation. We will simplify and solve for $x$ .

Combine like terms on the left side: $3 + 17 + x - 56x = 20 - 55x$ .
Equate the simplified expression to the right side: $20 - 55x = 18x + 44$ .
Move all terms involving $x$ to one side: $20 - 44 = 18x + 55x$ .
This simplifies to $-24 = 73x$ .
Solve for $x$ by dividing both sides by 73: $x = -\frac{24}{73}$ .

The solution for $x$ from the first equation is $x = -\frac{24}{73}$ .

Equation 2: $(x-2)(x+2) = 0$

This equation is a difference of squares, $(x^2 - 4) = 0$ .

Apply the zero-product property: $x - 2 = 0$ or $x + 2 = 0$ .
Solve each equation: $x = 2$ and $x = -2$ .

The solutions from the second equation are $x = 2$ and $x = -2$ .

Conclusion

From both equations, we have the possible values of $x$ as $-\frac{24}{73}$ , $-2$ , and $2$ .

The largest value among these is $2$ .

Therefore, the largest $x$ is $2$ .

3

Final Answer

2

Key Points to Remember

Essential concepts to master this topic

Linear Solving: Combine like terms, then isolate variable systematically
Zero Product: For (x-2)(x+2) = 0, set each factor to zero: x = 2, x = -2
Compare Solutions: Check -24/73 ≈ -0.33, -2, and 2 to find largest ✓

Common Mistakes

Avoid these frequent errors

Forgetting to compare all solutions from both equations
Don't just solve each equation separately without comparing = missing the final step! Students often solve correctly but forget to identify which x value is actually largest. Always list all solutions and compare their numerical values to find the maximum.

Practice Quiz

Test your knowledge with interactive questions

Solve:

$$ (2+x)(2-x)=0 $$

FAQ

Everything you need to know about this question

Why does the second equation have two solutions?

+

The equation (x-2)(x+2) = 0 is a product of two factors. When a product equals zero, at least one factor must be zero. So either x-2 = 0 (giving x = 2) or x+2 = 0 (giving x = -2).

How do I compare a fraction like -24/73 with whole numbers?

+

Convert the fraction to a decimal: -24/73 ≈ -0.33. Now you can easily compare: -2 < -0.33 < 2, so 2 is the largest.

Can I expand (x-2)(x+2) instead of using zero product property?

+

Yes! Expanding gives x² - 4 = 0, then x² = 4, so x = ±2. Both methods work, but zero product property is usually faster for factored equations.

What if I made an algebra mistake in the linear equation?

+

Always check your work! Substitute x = -24/73 back into the original equation 3 + x + 17 - 56x = 18x + 44. Both sides should give the same value.

Why do we need to find the largest x from all equations?

+

The question asks us to compare solutions from different equations. We're not solving a system (where x must satisfy both), but finding which individual solution has the greatest value.

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Solve and Compare: Tackle the Equations 3 + x + 17 - 56x = 18x + 44 and (x-2)(x+2) = 0

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