Question

Solve the inequality and graph the solution. Then check graphically whether the given \(x\) -value is a solution by graphing the \(x\) -value on the same number line. $$ -2 x \geq 6 \text { or } 2 x+1>5 ; x=0 $$

Short answer

The solutions to the inequalities are \( x \leq -3 \) and \( x > 2 \). The value \( x = 0 \) is not a solution to these inequalities.

Step-by-step solution

Solve the first inequality

The first inequality to solve is \( -2x \geq 6 \). Divide both sides of the equation by -2. Remember that when you divide by a negative number, the direction of the inequality sign changes. So, \( x \leq -3 \)

Solve the second inequality

The second inequality to solve is \( 2x + 1 > 5 \). Start by subtracting 1 from both sides to get \( 2x > 4 \). Then divide both sides by 2 and the result is \( x > 2 \)

Draw a number line graph

Draw a number line, marking all the numbers from -4 to 3. Circle -3 and everything to the left of it to represent \( x \leq -3 \). Then, circle the numbers greater than 2 up to 3 and beyond since \( x > 2 \). To indicate possible solutions, fill in -3 and optionally, make an open circle at 2 to show that it's not included.

Check the value \( x = 0 \)

Plot the value of \( x = 0 \) on the number line. If it falls within the range that was circled (which it doesn't), then it would be a solution, but since it falls outside, \( x = 0 \) is not a solution for the given inequalities.