|x+5|+9>10 solve for x
subtract 9 from each side
abs(x+5)>1
which means x must be greater than -4
|x+5| > 1
x > -4 or x < -6
Mark -5 on the number line
You want all the points more than 1 unit away from -5.
To solve the inequality |x+5| + 9 > 10 for x, we need to consider two cases, based on the absolute value expression.
Case 1: x+5 is positive or zero:
In this case, |x+5| = x+5. Substituting this into the inequality gives:
(x+5) + 9 > 10
Simplifying this, we get:
x + 14 > 10
Next, isolate x by subtracting 14 from both sides:
x > 10 - 14
This simplifies to:
x > -4
So, if x+5 is positive or zero, x must be greater than -4.
Case 2: x+5 is negative:
In this case, |x+5| = -(x+5). Substituting this into the inequality gives:
-(x+5) + 9 > 10
Simplifying this, we get:
- x - 5 + 9 > 10
Combine like terms:
- x + 4 > 10
Next, isolate x by subtracting 4 from both sides:
- x > 10 - 4
This simplifies to:
- x > 6
Now, multiply both sides of the inequality by -1 to change the direction of the inequality:
x < -6
So, if x+5 is negative, x must be less than -6.
Combining the results from both cases, we conclude that x can take any value that is less than -6 or greater than -4.
In interval notation, the solution is (-∞, -6) ∪ (-4, +∞).