Question

Solve the following inequality. Write the inequality in interval notation, and graph it 7(x+3)-8x≤ 3(2x + 1) -4x

Answer 1

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:

`7(x+3)-8x\leq3(2x+1)-4x`

First, we should apply the distributive property on both sides of the inequality:

`\begin{gathered} 7(x+3)\rightarrow7x+21 \\ \\ 3(2x+1)\rightarrow6x+3 \end{gathered}`

So we now have

`7x+21-8x\leq6x+3-4x`

Combing like terms on both sides, we get:

`\begin{gathered} (7x-8x)+21\leq(6x-4x)+3 \\ \\ -x+21\leq2x+3 \end{gathered}`

Add x to both sides:

`\begin{gathered} -x+x+21\leq2x+x+3 \\ \\ 21\leq3x+3 \end{gathered}`

Subtract 3 from both sides:

`\begin{gathered} 21-3\leq3x+3-3 \\ \\ 18\leq3x \end{gathered}`

Divide by 3 on both sides (the symbol will remain the same since 3 is positive):

`\begin{gathered} (18)/(3)\leq(3x)/(3) \\ \\ 6\leq x \end{gathered}`

We read the solution as x is greater than or equal to 6. In interval notation, we get:

`[6,\infty)`

On a graph, we have: