Chapter 1, Equations and Inequalities - 1.7 Solve Absolute Value Equations and Inequalities - 1.7 Exercises - Skill Practice - Page 55: 37

$x=-0.5$ or $x=-1.5$

$|4x+5|=2x+4$ implies $4x+5=2x+4\\2x=-1\\x=-0.5$ or $4x+5=-2x-4\\6x=-9\\x=-1.5$. Here we have no extraneous solutions. Thus $x=-0.5$ or $x=-1.5$ Check: $|4\cdot(-0.5)+5|=|-2+5|=3=-1+4=2\cdot-0.5+4$ and $|4\cdot(-1.5)+5|=|-6+5|=|-1|=1=-3+4=2\cdot-1.5+4$. Thus both are good solutions.