Answer
$$(1,3)$$
Work Step by Step
Let $x^2 - 4x + 3 \lt 0$ be a function $f$.
Find the $x$-intercepts by solving $x^2 - 4x + 3=0$
Factor: $x^2 - 4x + 3$ = $(x-1)(x-3)$
$x-1=0$ or $x-3=0$
$x = 1$ or $x = 3$
These x-intercepts serve as boundary points that separate the number line into intervals.
The boundary points of this equation therefore divide the number line into three intervals:
$(-∞, 1)(1,3)(3, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, 1)$
Test Value = $0$
$f=(x-1)(x-3)$
$f=(0-1)(0-3)$
$f=3$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, 1)$.
$(1,3)$
Test Value = $2$
$f=(x-1)(x-3)$
$f=(2-1)(2-3)$
$f=-1$
Conclusion: $f (x) < 0$ for all $x$ in $(1,3)$.
$(3,+∞)$
Test Value = $4$
$f= (x-1)(x-3)$
$f=(4-1)(4-3)$
$f=3$
Conclusion: $f (x) > 0$ for all $x$ in $(3,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) < 0$, where $f(x) = x^2 - 4x + 3$. Based on the solution above, $f(x)<0$ for all $x$ in $(1,3)$.
Thus, the solution set of the given inequality is:
$$(1,3)$$
