7.5 Linear Inequalities

7.5 – Linear Inequalities Goals / “I can…” Graph linear inequalities Write and use linear inequalities when modeling real – world situations

7.5 – Linear Inequalities RECALL Graph x > 4 on a number line. Graph y ≤ 2 on a number line. -4 -2 2 4 -4 -2 2 4

7.5 – Linear Inequalities REMEMBER: The type of dot on the number line is important. Open DOT means NOT INCLUDED (> or <) Closed DOT means INCLUDED (≤ or ≥)

Remember these symbols!!!! Less than Greater than Less than or EQUAL TO Greater than or EQUAL TO

7.5 – Linear Inequalities When graphing inequalities on the coordinate plane, we use a similar idea. Dashed lines mean the same as open circles. (> or <) Solid lines mean the same as closed circles. (≤ or ≥)

Graphing an Inequality in Two Variables Graph x < 2 Step 1: Start by graphing the line x = 2 Now what points would give you less than 2? Since it has to be x < 2 we shade everything to the left of the line.

7.5 – Linear Inequalities Oh, and did I mention we have to shade a part of the graph????? When considering shading, you shade the part of the graph that WORKS FOR THE EQUATION.

Graphing a Linear Inequality Sketch a graph of y  3

7.5 – Linear Inequalities Graph x ≥ 3

7.5 – Linear Inequalities Graph y ≤ 2x + 2

Solving an Inequality x < 8 Solve using addition: Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations. Solve using addition: x – 3 < 5 Add the same number to EACH side. +3 +3 x < 8

Solving Using Subtraction Subtract the same number from EACH side. -6 -6

THE TRAP….. When you multiply or divide each side of an inequality by a negative number, you must REVERSE the inequality SYMBOL to maintain a true statement.

Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. Multiply by (-1). (-1) See the switch

Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. -2

Solving using Multiplication Multiply each side by the same positive number. (2)

Some Helpful Hints If the sign is > or < the line is dashed If the sign is  or  the line will be solid When dealing with just x and y. If the sign > or  the shading either goes up or to the right If the sign is < or  the shading either goes down or to the left

Step 1: Put into slope intercept form Using What We Know Sketch a graph of x + y < 3 Step 1: Put into slope intercept form y < -x + 3 Step 2: Graph the line y = -x + 3

When dealing with slanted lines If it is > or  then you shade above If it is < or  then you shade below the line