To find the values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 3x + 5 \)[/tex] satisfies the inequality [tex]\( h(x) > 17 \)[/tex], follow these steps:
1. Set up the Inequality:
Given the function [tex]\( h(x) \)[/tex] and the inequality:
[tex]\[
h(x) > 17
\][/tex]
2. Substitute [tex]\( h(x) \)[/tex] into the Inequality:
Replace [tex]\( h(x) \)[/tex] with [tex]\( 3x + 5 \)[/tex]:
[tex]\[
3x + 5 > 17
\][/tex]
3. Isolate the Variable [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], start by isolating [tex]\( 3x \)[/tex] on one side of the inequality.
Subtract 5 from both sides:
[tex]\[
3x + 5 - 5 > 17 - 5
\][/tex]
Simplifying this yields:
[tex]\[
3x > 12
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides of the inequality by 3 to isolate [tex]\( x \)[/tex]:
[tex]\[
\frac{3x}{3} > \frac{12}{3}
\][/tex]
Simplifying this yields:
[tex]\[
x > 4
\][/tex]
5. Write the Final Solution:
The inequality [tex]\( 3x + 5 > 17 \)[/tex] holds true when:
[tex]\[
x > 4
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( h(x) > 17 \)[/tex] are those for which [tex]\( x \)[/tex] is greater than 4. In interval notation, this can be expressed as [tex]\( (4, \infty) \)[/tex].
