Inequalities and Absolute Value

Self-Test:

1) The solution set of the inequality $ x-3 < \frac{5}{x+1} $ is given by:

$(-2, 4)$

$(-\infty , -2) \cup (4, \infty)$

$(-\infty, -2) \cup (-1, 4)$

$(-2, 1) \cup (4, \infty)$

All $x \neq -1$.

Hint Move everything to one side, but be careful that $(x+1)$ might be negative.

2) Which picture represents the interval $1\leq x < 3$?

Figure generated by Wolfram|Alpha on Sep. 3, 2013.

Hint Try expressing the intervals in the picture using inequalities.

3) The solution set of the inequality $0< | x-1 | < r$ consists of:

All points within distance $1$ of $r$.

All points within distance $r$ of $1$.

$(1-r, 1) \cup (1, 1+r)$.

$(1-r, 1+r)$

None of the above

Hint Be careful of the double inequality. Try sketching both inequalities separately, then taking a common solution.

4) Which picture represents the set of points $(x,y)$ that satisfy $2x-5y>10$:
Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{2}{5} x -2$	Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{2}{5} x -2$
Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{5}{2} x -5$	Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{5}{2} x -5$
None of the above

Hint Be mindful of the signs of the coefficients.

5) Which of the following describes precisely the values of $x$ that are between $-1$ and $5$ inclusive or are strictly greater than $9$?

$(-1,5)\cup(9,+\infty)$

$(-1,5)\cup[9,+\infty)$

$[-1,5)\cup(9,+\infty)$

$(-1,5]\cup(9,+\infty)$

$[-1,5]\cup(9,+\infty)$

Hint Start by sketching the points on the number line.

6) The solution to $2x+3<5x+12$ is:

$x<-3$

$x>-3$

$x<\frac{15}{7}$

$x<-5$

$x>-5$

Hint Isolate the terms with $x$.

7) If you shade the region of the plane described by $y \geq 3x + 5$, then the points $( 0 , 0 )$ and $( -1 , 1 )$

are both located in the shaded region.

are both located in the non-shaded region

are situated so that $( 0 , 0 )$ is located in the shaded region, but $( -1 , 1 )$ is not.

are situated so that $( 0 , 0 )$ is located in the non-shaded region, but $( -1 , 1 )$ is not.

can not be located based on the information given.

Hint Draw the picture.

8) All of the solutions to $|x-2| + 2x \leq 16$ can be described by:

$2\leq x \leq 6$ and $x \geq 14$

$x \leq 2$ and $6 \leq x \leq 14$

$-6 \leq x \leq 14$

$x \leq 6$

$x \geq 14$

Hint Don't forget that an inequality with absolute value is really two inequalities.

9) $ x^3 + x^2 - 2x > 0 $ for all values of x in

$(1, \infty)$

$(-\infty,2)$

$(-2,0)\cup (1,\infty)$

$(-\infty,-2)\cup (0,1)$

$(-2,1)$

Hint Begin by factoring.

10) $|x-1|<|x-3|$ for all values of $x$ in

$(1,3)$

$(-\infty,1)\cup (3,\infty)$

$(-2,\infty)$

$(2,\infty)$

$(-\infty, 2)$

Hint Graphs will help here.

4) Which picture represents the set of points $(x,y)$ that satisfy $2x-5y>10$:
Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{2}{5} x -2$	Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{2}{5} x -2$
Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{5}{2} x -5$	Figure generated by Wolfram\|Alpha on Sep. 3, 2013. line: $y=\frac{5}{2} x -5$
None of the above

Inequalities and Absolute Value

Self-Test:

Worked Examples and Practice Problems

Worked Examples

Practice Problems