In this problem, we need to solve and graph a linear inequality. We can begin by solving it like a regular equation, but we have to be careful with the last step.
If we ever multiply or divide by a negative number in the inequality, the symbol will switch directions.
Let's get started.
We are given:
[tex]7(x+3)-8x\leq3(2x+1)-4x[/tex]First, we should apply the distributive property on both sides of the inequality:
[tex]\begin{gathered} 7(x+3)\rightarrow7x+21 \\ \\ 3(2x+1)\rightarrow6x+3 \end{gathered}[/tex]So we now have
[tex]7x+21-8x\leq6x+3-4x[/tex]Combing like terms on both sides, we get:
[tex]\begin{gathered} (7x-8x)+21\leq(6x-4x)+3 \\ \\ -x+21\leq2x+3 \end{gathered}[/tex]Add x to both sides:
[tex]\begin{gathered} -x+x+21\leq2x+x+3 \\ \\ 21\leq3x+3 \end{gathered}[/tex]Subtract 3 from both sides:
[tex]\begin{gathered} 21-3\leq3x+3-3 \\ \\ 18\leq3x \end{gathered}[/tex]Divide by 3 on both sides (the symbol will remain the same since 3 is positive):
[tex]\begin{gathered} \frac{18}{3}\leq\frac{3x}{3} \\ \\ 6\leq x \end{gathered}[/tex]We read the solution as x is greater than or equal to 6. In interval notation, we get:
[tex][6,\infty)[/tex]On a graph, we have:
