Question

Solve the inequality and graph the solution. $$ 2 x+7<3 \text { or } 5 x+5 \geq 10 $$

Short answer

The solution to the inequalities are \(x < -2\) or \(x \geq 1\).

Step-by-step solution

Solve the first inequality

Start by rearranging the first inequality, \(2x + 7 < 3\), to solve for \(x\). To do this, first subtract 7 from both sides of the inequality to get \(2x < -4\). Then, divide both sides by 2 to solve for \(x\), which gives you \(x < -2\). That's the solution for the first inequality.

Solve the second inequality

Next, solve the second inequality, \(5x + 5 \geq 10\). First, subtract 5 from both sides of the inequality, which gives you \(5x \geq 5\). Then, divide both sides by 5 to solve for \(x\), giving \(x \geq 1\). That's the solution for the second inequality.

Graph the solution

To graph the solution, start by marking the solutions for each inequality on the number line. For \(x < -2\), plot an open circle at -2 and draw a line extending to the left to represent all numbers less than -2. For \(x \geq 1\), plot a closed circle at 1, and draw a line extending to the right to represent all numbers greater than or equal to 1. Since the problem statement said 'or', the solution to the original problem is the union of the two solutions, meaning any number that is part of either solution set is a solution.