Solving g(x)=-x+4 > 0: Linear Function Inequality Analysis

Q: Why does the inequality flip when I multiply by -1?

Think of it this way: if -2 > -5, then multiplying by -1 gives us 2 . The inequality direction must flip to keep the statement true! This is a fundamental rule in algebra.

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

Look where the line $ g(x) = -x + 4 $ is above the x-axis (positive values). You can see this happens when \( x , which confirms our algebraic solution!

Q: What does g(x) > 0 mean visually?

$ g(x) > 0 $ means we want the y-values of the function to be positive. On a graph, this is everywhere the line sits above the x-axis.

Q: Why is x = 4 not included in the solution?

At $ x = 4 $, we have $ g(4) = -4 + 4 = 0 $. Since we need g(x) > 0 (strictly greater than), zero doesn't count. That's why we use $ x instead of \( x ≤ 4 $.

Q: How do I know which direction the line slopes?

In $ g(x) = -x + 4 $, the coefficient of x is -1 (negative), so the line slopes downward from left to right. This means it starts high and goes low as x increases.

Linear Inequalities with Graphical Interpretation

Look at the graph below of the following functions:

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

$g(x)=-x+4$

For which values of x is
$g(x)>0$ true?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 For which values is the function positive?

00:04 We want to find the intersection point with the X-axis

00:07 Let's isolate X

00:11 This is the intersection point, above which the function is positive

00:14 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the graph below of the following functions:

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

$g(x)=-x+4$

For which values of x is
$g(x)>0$ true?

Step-by-step solution

To solve the problem of finding for which values of $x$ , the function $g(x) = -x + 4$ is greater than zero, we begin as follows:

Step 1: Set up the inequality $g(x) > 0$ . This translates to $-x + 4 > 0$ .
Step 2: Solve the inequality:
- Subtract 4 from both sides: $-x > -4$ .
- Multiply both sides by $-1$ , reversing the inequality: $x < 4$ .

Therefore, the solution to the problem is that $g(x) > 0$ when $x < 4$ .

The corresponding choice that reflects this solution is choice 4: $x < 4$ .

Final Answer

$x<4$

Key Points to Remember

Essential concepts to master this topic

Linear Inequality: Solve $-x + 4 > 0$ by isolating x
Inequality Reversal: When multiplying by negative one: $-x > -4$ becomes $x < 4$
Graph Check: Function $g(x) = -x + 4$ is above x-axis when $x < 4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to reverse inequality when multiplying by negative
Don't keep the same inequality direction when multiplying $-x > -4$ by -1 = wrong answer $x > 4$ ! This gives the opposite region where the function is negative. Always flip the inequality sign when multiplying or dividing by a negative number.

Practice Quiz

Test your knowledge with interactive questions

Which formula describes graph 2?

FAQ

Everything you need to know about this question

Why does the inequality flip when I multiply by -1?

Think of it this way: if -2 > -5, then multiplying by -1 gives us 2 < 5. The inequality direction must flip to keep the statement true! This is a fundamental rule in algebra.

How can I check my answer using the graph?

Look where the line $g(x) = -x + 4$ is above the x-axis (positive values). You can see this happens when $x < 4$ , which confirms our algebraic solution!

What does g(x) > 0 mean visually?

$g(x) > 0$ means we want the y-values of the function to be positive. On a graph, this is everywhere the line sits above the x-axis.

Why is x = 4 not included in the solution?

At $x = 4$ , we have $g(4) = -4 + 4 = 0$ . Since we need g(x) > 0 (strictly greater than), zero doesn't count. That's why we use $x < 4$ instead of $x ≤ 4$ .

How do I know which direction the line slopes?

In $g(x) = -x + 4$ , the coefficient of x is -1 (negative), so the line slopes downward from left to right. This means it starts high and goes low as x increases.

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