Solve f(x)=x²-6x+8 > -x+4: Quadratic-Linear Function Comparison

Q: Why do I need to rearrange to get everything on one side?

Moving everything to one side creates a standard inequality form that's easier to factor and analyze. This lets you find where the expression equals zero, which are the boundary points for your solution intervals.

Q: How do I know which intervals to include in my answer?

After factoring, test one point from each interval in your inequality. If the test makes the inequality true, include that interval. If false, exclude it from your solution.

Q: What's the difference between the graph method and algebraic method?

The graph shows where f(x) is above g(x) visually, while algebra gives the exact answer. Both should give the same result - use the graph to check your algebraic work!

Q: Why are the boundary points x = 1 and x = 4 not included?

The inequality uses > (greater than), not ≥. At x = 1 and x = 4, f(x) = g(x), so f(x) is not greater than g(x). Use open intervals: x 4.

Q: How can I check my answer using the graph?

Look where the parabola (blue curve) is above the line (gray). This happens for x-values to the left of point C and to the right of point B, confirming x 4.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 For which values is the parabola greater than the line?

00:03 Let's compare the functions to find their intersection points

00:09 Let's arrange the equation so that one side equals 0

00:17 Let's use the quadratic formulas

00:23 These are the intersection points

00:27 Using the line, we'll find the domains where the parabola is greater

00:44 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

The following function is graphed below:

$f(x)=x^2-6x+8$

$g(x)=-x+4$

For which values of x is
$f(x) > g(x)$ true?

2

Step-by-step solution

To determine for which values of $x$ the condition $f(x) > g(x)$ holds, follow these steps:

Step 1: Set up the inequality $x^2 - 6x + 8 > -x + 4$ .
Step 2: Rearrange all terms to one side: $x^2 - 6x + 8 + x - 4 > 0$ simplifies to $x^2 - 5x + 4 > 0$ .
Step 3: Solve the related quadratic equation $x^2 - 5x + 4 = 0$ . This factors as $(x - 1)(x - 4) = 0$ , giving roots $x = 1$ and $x = 4$ .
Step 4: Determine intervals defined by the roots: $(-\infty, 1)$ , $(1, 4)$ , and $(4, \infty)$ .
Step 5: Test points from each interval in the inequality $x^2 - 5x + 4 > 0$ :
- For $x = 0$ (from interval $(-\infty, 1)$ ), $x^2 - 5x + 4 = 4$ which is greater than 0.
- For $x = 2$ (from interval $(1, 4)$ ), $x^2 - 5x + 4 = -2$ which is less than 0.
- For $x = 5$ (from interval $(4, \infty)$ ), $x^2 - 5x + 4 = 9$ which is greater than 0.
Step 6: From the test points, determine $f(x) > g(x)$ in intervals $(-\infty, 1)$ and $(4, \infty)$ .

Thus, $f(x) > g(x)$ for $x < 1$ or $x > 4$ .

Therefore, the solution to the problem is $x < 1, 4 < x$ , corresponding to choice 2.

3

Final Answer

$x < 1,4 < x$

Key Points to Remember

Essential concepts to master this topic

Setup: Convert f(x) > g(x) to standard form by moving all terms to one side
Technique: Factor x²-5x+4 = (x-1)(x-4) = 0 to find critical points x = 1, 4
Check: Test intervals with sign analysis: x = 0 gives +4 > 0, x = 2 gives -2 < 0 ✓

Common Mistakes

Avoid these frequent errors

Testing only one point or misreading the inequality direction
Don't just find where the functions intersect and guess the answer = wrong intervals! Students often forget that > means "above" on the graph. Always test a point from each interval in your factored inequality and check which intervals make it positive.

Practice Quiz

Test your knowledge with interactive questions

The following functions are graphed below:

$$ f(x)=x^2-6x+8 $$

$$ g(x)=4x-17 $$

For which values of x is
$$ f(x)<0 $$ true?

FAQ

Everything you need to know about this question

Why do I need to rearrange to get everything on one side?

+

Moving everything to one side creates a standard inequality form that's easier to factor and analyze. This lets you find where the expression equals zero, which are the boundary points for your solution intervals.

How do I know which intervals to include in my answer?

+

After factoring, test one point from each interval in your inequality. If the test makes the inequality true, include that interval. If false, exclude it from your solution.

What's the difference between the graph method and algebraic method?

+

The graph shows where f(x) is above g(x) visually, while algebra gives the exact answer. Both should give the same result - use the graph to check your algebraic work!

Why are the boundary points x = 1 and x = 4 not included?

+

The inequality uses > (greater than), not ≥. At x = 1 and x = 4, f(x) = g(x), so f(x) is not greater than g(x). Use open intervals: x < 1 or x > 4.

How can I check my answer using the graph?

+

Look where the parabola (blue curve) is above the line (gray). This happens for x-values to the left of point C and to the right of point B, confirming x < 1 and x > 4.

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Solve f(x)=x²-6x+8 > -x+4: Quadratic-Linear Function Comparison

Quadratic Inequality with Graphical Analysis

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