Math Study Guide for the SHSAT

Expressions, Equations, and Inequalities

In mathematics, expressions, equations, and inequalities are fundamental concepts used to describe relationships between quantities. Whether you’re dealing with linear expressions, problem-solving with signed numbers, or solving word problems using variables, understanding these concepts is necessary for various mathematical tasks.

Linear Expressions

Linear expressions involve terms that are either constants or products of constants and variables raised to the first power (raised to the power of \(1\)). Linear expressions can be manipulated in various ways, including addition, subtraction, factoring, expanding, and writing them in different forms to depict relationships between quantities.

Adding and Subtracting

Adding and subtracting linear expressions involves combining like terms to simplify the expression. Like terms are terms with the same variables raised to the same powers.

Consider the linear expressions \(3x+2\) and \(4x-5\).

To add these two expressions, we have to add the like terms. In this case, \(3x\) and \(4x\) are like terms, and \(2\) and \(-5\) are like terms. So, let’s group the like terms together and add:

\[(3x+2)+(4x-5)\] \[=3x+4x+2-5\] \[=7x-3\]

Factoring

Factoring linear expressions involves expressing them as products of simpler expressions. Factoring can help when simplifying expressions and solving equations. This process generally involves finding the greatest common factor (GCF) of the terms within an expression.

An example will make this clearer.

Factor the expression \(6x+12\).

To factor this expression, we need to find the GCF between the two terms. In this case, that is \(6\):

\[6x+12\] \[=6 \times x + 6 \times 2\] \[=6(x+2)\]

Expanding

Expanding linear expressions involves distributing terms to remove parentheses and simplify the expression. This is the reverse process of factoring. Let’s look at an example.

Expand the expression \(3(x+4)\).

\[3(x+4)\] \[=3 \times x + 3 \times 4\] \[=3x+12\]

Writing in Different Forms

Linear expressions can be written in different forms to represent various relationships between quantities. These forms include the standard form, slope-intercept form, and point-slope form.

This is the standard form of a linear equation:

\[2x-3y=6\]

If we wanted to see how quickly \(y\) is changing with respect to \(x\), we would change it to the slope-intercept form, which looks like this:

\[y = \frac{2}{3}x - 2\]

To use the same equation to see the slope of the line and a point that the line passes through at the same time, we would change it to the point-slope form, like this:

\[y - 2 = \frac{2}{3}(x - 6)\]

Solving Word Problems Using Variables

Word problems often involve unknown quantities that can be represented by variables. Solving these problems requires translating the information provided into mathematical expressions, equations, or inequalities. This process is involved in all types of real-world problems and will be useful in various career fields, including finance, physics, and engineering.

Using Equations

Equations are mathematical statements asserting that two expressions are equal. They are used to represent relationships between quantities and are solved to find the value of the variable.

Suppose you are told the sum of two unknown numbers is \(10\) and one of the numbers is three times the other. To figure out what the unknown numbers are, you can use variables to set up an equation.

Let \(x\) represent one unknown number and \(y\) represent the other number. We can write the following equations to represent the first relationship:

\[x+y=10\]

We also know that one of the numbers is three times the other, so we can say this:

\[y=3x\]

With this information, we can figure out the values of the two variables by substituting a value from one equation into the other. For instance, based on the second equation, we can substitute \(3x\) for \(y\) in the first equation:

\[x+3x=10\] \[4x=10\] \[x=\frac{10}{4}\] \[x=2.5\]

Now, we can find \(y\) using the second equation:

\[y=3x\] \[y=3 \times 2.5\] \[=7.5\]

So, the smaller number, \(x\), is \(2.5\), and the larger number, \(y\), is \(7.5\).

Using Inequalities

Inequalities compare two expressions and show their relationship. They are used when the relationship between quantities is not equal. There are four types of inequality, and they are expressed with common symbols: less than (\(<\)), greater than (\(>\)), less than or equal to (\(\le\)), and greater than or equal to (\(\ge\)).

Consider a scenario where you earn \(\$10\) per hour and want to determine the number of hours you need to work to earn at least \(\$100\). Let \(x\) represent the number of hours you work and \(y\) represent your earnings. The inequality representing this relationship would be:

\[10x \geq y\]

To find the range of values for \(x\), we must solve this inequality:

\[10x\geq 100\] \[x\geq \frac{100}{10}\] \[x\geq 10\]

So, you would need to work at least \(10\) hours to earn at least \(\$100\).

Graphing Inequalities on a Number Line

Graphing inequalities on a number line provides a visual representation of the solutions to the inequality, which can be useful for understanding and solving problems. Consider this example problem.

Graph the inequality \(2x+3 \geq 7\) on a number line to illustrate the solution set.

\[2x+3 \geq 7\] \[2x \geq 7-3\] \[2x \geq 4\] \[x \geq2\]

If the graphed range includes the endpoint number (\(\geq\) or \(\leq\)), its point is filled in (solid). If the range does not include the endpoint number (\(>\) or \(<\)), the point is not filled in (hollow).