In Exercises 105–106, use the table to solve each inequality. - 3 < 2x - 5 ≤ 3

Verified step by step guidance

1. Step 1: Start with the given inequality: -3 < 2x - 5 ≤ 3.
2. Step 2: Add 5 to all parts of the inequality to isolate the term with x: -3 + 5 < 2x - 5 + 5 ≤ 3 + 5.
3. Step 3: Simplify the inequality: 2 < 2x ≤ 8.
4. Step 4: Divide all parts of the inequality by 2 to solve for x: 2/2 < 2x/2 ≤ 8/2.
5. Step 5: Simplify the inequality: 1 < x ≤ 4.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can be represented using symbols such as <, >, ≤, and ≥. In this case, the inequality 3 < 2x - 5 ≤ 3 indicates that we need to find the values of x that satisfy both conditions simultaneously.

Linear Functions

A linear function is a function that can be graphically represented as a straight line. It is typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. The equation y = 2x - 13 represents a linear function, and understanding its graph helps in solving the given inequality.

Solving for x

Solving for x involves isolating the variable x in an equation or inequality to find its possible values. In the context of the given inequality, we will manipulate the expression 2x - 5 to determine the range of x that satisfies the conditions set by the inequality. This often includes adding, subtracting, multiplying, or dividing both sides of the inequality.