Question

name date consider the equation |2x + 5| = 3. a. rewrite the equation as a compound statement. b. use the compound statement to find the solution set of the original equation. c. use the original equation to verify the solutions found in part (b). d. explain why the equation |2x + 5| = -3 has no solution.

Explanation:

Part a

Step1: Apply absolute value rule

For $|A|=b$ where $b>0$, $A=b$ or $A=-b$.
So $2x+5=3$ or $2x+5=-3$

Part b

Step1: Solve first equation

Isolate $x$ in $2x+5=3$
$2x=3-5$
$2x=-2$
$x=\frac{-2}{2}=-1$

Step2: Solve second equation

Isolate $x$ in $2x+5=-3$
$2x=-3-5$
$2x=-8$
$x=\frac{-8}{2}=-4$

Part c

Step1: Verify $x=-1$

Substitute $x=-1$ into $|2x+5|$:
$|2(-1)+5|=|-2+5|=|3|=3$

Step2: Verify $x=-4$

Substitute $x=-4$ into $|2x+5|$:
$|2(-4)+5|=|-8+5|=|-3|=3$

Part d

Step1: Recall absolute value property

The absolute value of any real number is non-negative, meaning $|A|\geq0$ for all real $A$.

Answer:

a. $2x + 5 = 3$ or $2x + 5 = -3$
b. $\{-4, -1\}$
c. Both $x=-1$ and $x=-4$ satisfy the original equation, as substituting them results in $3$, which matches the right-hand side.
d. The absolute value of any expression is always greater than or equal to 0, so it can never equal a negative number like $-3$. Thus, $|2x + 5| = -3$ has no solution.