Question

an inequality is shown.

$-3x + 5 < 2 \left( x - 10 \
ight)$

solve the inequality and graph the solution on the number line.

Explanation:

Step1: Expand the right side

First, we expand the right - hand side of the inequality \(-3x + 5\lt2(x - 10)\). Using the distributive property \(a(b + c)=ab+ac\) (here \(a = 2\), \(b=x\), \(c=- 10\)), we get \(2(x - 10)=2x-20\). So the inequality becomes \(-3x + 5\lt2x-20\).

Step2: Move the x - terms to one side

We want to get all the \(x\) - terms on one side. Let's add \(3x\) to both sides of the inequality. Adding \(3x\) to both sides gives \(-3x+3x + 5\lt2x + 3x-20\), which simplifies to \(5\lt5x-20\).

Step3: Move the constant term to the other side

Next, we add 20 to both sides of the inequality \(5\lt5x-20\). Adding 20 to both sides, we have \(5 + 20\lt5x-20 + 20\), which simplifies to \(25\lt5x\).

Step4: Solve for x

Now, we divide both sides of the inequality \(25\lt5x\) by 5. Dividing both sides by 5 (since 5 is positive, the direction of the inequality sign remains the same), we get \(\frac{25}{5}\lt\frac{5x}{5}\), which simplifies to \(5\lt x\) or \(x\gt5\).

(For graphing: On the number line, we draw an open circle at \(x = 5\) (because the inequality is strict, \(x
eq5\)) and then draw an arrow to the right of 5 to represent all the numbers greater than 5.)

Answer:

The solution to the inequality is \(x\gt5\). For the graph, there is an open circle at \(x = 5\) and the line is shaded to the right of 5.