2.7 Linear inequalities and absolute value inequalities  (Page 2/11)

We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at $-\infty$ and ends at $\mathrm{-1},$ which is written as $\left(-\infty ,\mathrm{-1}\right].$

The second interval must show all real numbers greater than or equal to $1,$ which is written as $\left[1,\infty \right).$ However, we want to combine these two sets. We accomplish this by inserting the union symbol, $\cup ,$ between the two intervals.

The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.

1. 
   $\begin{array}{ll}x-15<4\hfill & \hfill \\ x-15+15<4+15 \hfill & \text{Add 15 to both sides}.\hfill \\ x<19\hfill & \hfill \end{array}$
2. 
   $\begin{array}{ll}6\ge x-1\hfill & \hfill \\ 6+1\ge x-1+1\hfill & \text{Add 1 to both sides}.\hfill \\ 7\ge x\hfill & \hfill \end{array}$
3. 
   $\begin{array}{ll}x+7>9\hfill & \hfill \\ x+7-7>9-7\hfill & \text{Subtract 7 from both sides}.\hfill \\ x>2\hfill & \hfill \end{array}$

1. 
   $\begin{array}{l}3x<6\hfill \\ \frac{1}{3}(3x)<(6)\frac{1}{3}\hfill \\ x<2\hfill \end{array}$
2. 
   $\begin{array}{ll}-2x-1\ge 5\hfill & \hfill \\ -2x\ge 6\hfill & \hfill \\ \left(-\frac{1}{2}\right)(-2x)\ge (6)\left(-\frac{1}{2}\right)\hfill & \text{Multiply by }-\frac{1}{2}.\hfill \\ x\le -3\hfill & \text{Reverse the inequality}.\hfill \end{array}$
3. 
   $\begin{array}{ll}5-x>10\hfill & \hfill \\ -x>5\hfill & \hfill \\ (-1)(-x)>(5)(-1)\hfill & \text{Multiply by }-1.\hfill \\ x<-5\hfill & \text{Reverse the inequality}.\hfill \end{array}$

Solving this inequality is similar to solving an equation up until the last step.

The solution set is given by the interval $\left(-\infty ,1\right],$ or all real numbers less than and including 1.

We begin solving in the same way we do when solving an equation.

The solution set is the interval $\left(-\infty ,\frac{15}{34}\right].$